arXiv subjects

Ben Li

Publications and source records attributed to Ben Li.

25 records · Page 2Linked to original sources

arXiv2019

The Loewner function of a log-concave function

We introduce the notion of Loewner (ellipsoid) function for a log concave function and show that it is an extension of the Loewner ellipsoid for convex bodies. We investigate its duality relation to the recently defined John (ellipsoid) function by Alonso-Gutierrez, Merino, Jimenez and Villa. For convex bodies, John and Loewner ellipsoids are dual to each other. Interestingly, this need not be the case for the John function and the Loewner function.

math.FA↗

arXiv2018

Floating functions

We introduce floating bodies for convex, not necessarily bounded subsets of $\mathbb{R}^n$. This allows us to define floating functions for convex and log concave functions and log concave measures. We establish the asymptotic behavior of the integral difference of a log concave function and its floating function. This gives rise to a new affine invariant which bears striking similarities to the Euclidean affine surface area.

math.FA↗

arXiv2017

Exact values of quantum violations in low-dimensional Bell correlation inequalities

The famous Clauser-Horne-Shimony-Holt (CHSH) inequality certifies a quantum violation, by a factor $\sqrt{2}$, of correlations predicted by the classical view of the world in the simplest possible nontrivial measurement setup (two systems with two dichotomic measurements each). In such setting, this is the largest possible violation, which is known as the \emph{Tsirelson bound}. In this paper we calculate the exact values of quantum violations for the other Bell correlation inequalities that appear in the setups involving up to four measurements; they are all smaller than $\sqrt{2}$. While various authors investigated these inequalities via numerical methods, our approach is analytic. We also include tables summarizing facial structure of Bell polytopes in low dimensions.

quant-ph↗

arXiv2016

Exotic Multi-fold Vortex Lattices of Spin-Orbit Coupled Bose-Einstein Condensates in Optical Lattices

We investigate the ground state of two-dimensional Bose-Einstein condensates with Rashba spin-orbit coupling in square optical lattices and demonstrate the existence of rich phases with different lattice structures, which is closely related to the degenerate structure of single particle energy spectrum induced by the competition of spin-orbit coupling and optical lattices. We find that the ground state is in the phase with either parity-time-reversal or parity symmetry by direct numerical simulation. We show the phase diagram of ground state in the whole regime of spin-orbit coupling strength, and particularly find that the system supports multi-fold vortex lattices, in which ground state holds half-quantum vortex lattices, vortex-antivortex pair lattices and fundamental vortex lattices, simultaneously, when single particle energy minimums touch the boundary of the first Brillouin zone.

cond-mat.quant-gas↗

arXiv2015

Bound on Bell Inequalities by Fraction of Determinism and Reverse Triangle Inequality

It is an established fact that entanglement is a resource. Sharing an entangled state leads to non-local correlations and to violations of Bell inequalities. Such non-local correlations illustrate the advantage of quantum resources over classical resources. Here, we study quantitatively Bell inequalities with $2\times n$ inputs. As found in [N. Gisin et al., Int. J. Q. Inf. 5, 525 (2007)] quantum mechanical correlations cannot reach the algebraic bound for such inequalities. In this paper, we uncover the heart of this effect which we call the {\it fraction of determinism}. We show that any quantum statistics with two parties and $2 \times n$ inputs exhibits nonzero fraction of determinism, and we supply a quantitative bound for it. We then apply it to provide an explicit {\it universal upper bound} for Bell inequalities with $2\times n$ inputs. As our main mathematical tool we introduce and prove a {\it reverse triangle inequality}, stating in a quantitative way that if some states are far away from a given state, then their mixture is also. The inequality is crucial in deriving the lower bound for the fraction of determinism, but is also of interest on its own.

quant-ph↗

arXiv2014

The creation of two-dimensional composite solitons in spin-orbit-coupled self-attractive Bose-Einstein condensates in free space

It is commonly known that two-dimensional mean-field models of optical and matter waves with the cubic self-attraction cannot produce stable solitons in free space because of the occurrence of the collapse in the same setting. By means of the numerical analysis and variational approximation, we demonstrate that the two-component model of the Bose-Einstein condensate, with the spin-orbit Rashba coupling and cubic attractive interactions, gives rise to solitary-vortex complexes of two types: semi-vortices (SVs, with a vortex in one component and a fundamental soliton in the other), and mixed modes (MMs, with topological charges 0 and +1/-1 mixed in both components). These two- dimensional composite modes can be created using the trapping harmonic-oscillator (HO) potential, but remain stable in the free space, if the trap is gradually removed. The SVs and MMs realize the ground state of the system, provided that the self-attraction in the two components is, respectively, stronger or weaker than the cross-attraction between them. The SVs and MMs which are not the ground state are subject to a drift instability. In the free space (in the absence of the HO trap), modes of both types degenerate into unstable Townes solitons when their norms attain the respective critical values, while there is no lower existence threshold for the stable modes. Moving free-space stable solitons are also found in the present non-Galilean-invariant system, up to a critical velocity. Collisions between two moving solitons lead to their merger into a single one.

cond-mat.quant-gas↗

arXiv2013

Vortex lattice solutions to the Gross-Pitaevskii equation with spin-orbit coupling in optical lattices

Effective spin-orbit coupling can be created in cold atom systems using atom-light interaction. We study the BECs in an optical lattice using the Gross-Pitaevskii equation with spin-orbit coupling. Bloch states for the linear equation are numerically obtained, and compared with stationary solutions to the Gross-Pitaevskii equation with nonlinear terms. Various vortex lattice states are found when the spin-orbit coupling is strong.

cond-mat.quant-gas↗