Localization and recurrence of a quantum walk in a periodic potential on a line

We present a numerical study of a model of quantum walk in a periodic potential on a line. We take the simple view that different potentials have different affects on the way in which the coin state of the walker is changed. For simplicity and definiteness, we assume that the walker＇s coin state is unaffected at sites without the potential, and rotated in an unbiased way according to the Hadamard matrix at sites with the potential. This is the simplest and most natural model of a quantum walk in a periodic potential with two coins. Six generic cases of such quantum walks are studied numerically. It is found that, of the six cases, four cases display significant localization effect where the walker is confined in the neighborhood of the origin for a sufficiently long time. Associated with such a localization effect is the recurrence of the probability of the walker returning to the neighborhood of the origin.

Spin–orbit-coupled spin-1 Bose–Einstein condensates confined in radially periodic potential

We investigate the ground states of spin-1 Bose–Einstein condensates (BECs) with spin–orbit coupling in a radiallyperiodic potential by numerically solving the coupled Gross–Pitaevskii equations. In the radially periodic potential, wefirst demonstrate that spin–orbit-coupled antiferromagnetic BECs support a multiring petal phase. Polar–core vortex canbe observed from phase profiles, which is manifested as circularly symmetric distribution. We further show that spin–orbitcoupling can induce multiring soliton structure in ferromagnetic BECs. It is confirmed especially that the wave-functionphase of the ring corresponding to uniform distribution satisfies the rotational symmetry, and the wave-function phase ofthe ring corresponding to partial splitting breaks the rotational symmetry. Adjusting the spin–orbit coupling strength cancontrol the number of petal in antiferromagnetic BECs and the winding numbers of wave-function in ferromagnetic BECs.Finally, we discuss effects of spin-independent and spin-dependent interactions on the ground states.

Gap solitons of spin–orbit-coupled Bose–Einstein condensates in PT periodic potential

We numerically investigate the gap solitons in Bose–Einstein condensates(BECs)with spin–orbit coupling(SOC)in the parity–time(PT)-symmetric periodic potential.We find that the depths and periods of the imaginary lattice have an important influence on the shape and stability of these single-peak gap solitons and double-peak gap solitons in the first band gap.The dynamics of these gap solitons are checked by the split-time-step Crank–Nicolson method.It is proved that the depths of the imaginary part of the PT-symmetric periodic potential gradually increase,and the gap solitons become unstable.But the different periods of imaginary part hardly affect the stability of the gap solitons in the corresponding parameter interval.

PERIODIC SOLUTIONS FOR HAMILTONIANSYSTEMS IN A NONCONVEX POTENTIAL WELL

Recently there has been a considerable amount of work on the existence of Tperiodic solutions for Hamiltonian systems with singular potentials, (see [1]—[7],[10], [11], [13], [14]). In this paper we will study the existence of T-periodic solutions for nonconservative second-order dynamical systems

A Simple Model for the Calculation of Diffusion Coefficient in a Periodic Potential

Motivated by developing a simple model to calculate the diffusion coefficient in moderate friction region, a simplified model is proposed to deal with the diffusion of Brownian particles in a periodic potential. Where the internal noise is a Gaussian white noise, and the basic cell of the periodic potential is composed of a parabolic potential linked with a harmonic potential. When the particles cross the joint point of the potential, a time coarse-graining scheme is used to obtain a simple analytical expression of the probability distribution. The particles drift and diffuse from the first barrier to the second barrier, the passing probability over the second barrier corresponding to the escape rate becomes decrease serves as the long-jump probability. The theoretical result is confirmed by numerical simulation results. The approach can be extended to color noise case.

Topological phase transitions and Majorana zero modes induced by the periodic potential in an antiferromagnetic chain

We investigate the topological properties of an antiferromagnetic(AFM)chain with an on-site periodic potential,considering the intrinsic spin–orbit coupling and an external Zeeman field along with the nanowire.Our results indicate that Majorana zero modes(MZMs)can be observed by adjusting the strength of the periodic potential.We have calculated the energy spectrum,the wave-function and transport properties,and all these results support the existence of MZMs in the AFM chain.Additionally,multiple topological phase transitions occur as the strength of the periodic potential changes,and several regions support MZMs.

Potential risks of organic contaminated soil after persulfate remediation:Harmful gaseous sulfur release

Persulfate is considered a convenient and efficient remediation agent for organic contaminated soil.However,the potential risk of sulfur into the soil remediation by persulfate remains ignored.In this study,glass bottles with different persulfate dosages and groundwater tables were set up to simulate persulfate remediation of organic pollutants(aniline).The results found sulfate to be the main end-product(83.0%–99.5%)of persulfate remediation after10 days.Moreover,H_(2)S accounted for 93.4%–99.4%of sulfur reduction end-products,suggesting that H_(2)S was the final fate of sulfur.H_(2)S was released rapidly after one to three days at a maximum concentration of 33.0 ppm,which is sufficient to make a person uncomfortable.According to the fitted curve results,H_(2)S concentration decreased to a safe concentration(0.15 ppm)after 20–85 days.Meanwhile,the maximum concentration of methanethiol reached 0.6 ppm.These results indicated that secondary pollution from persulfate remediation could release harmful gases over a long time.Therefore,persulfate should be used more carefully as a remediation agent for soil contamination.

Numerical Solution for a Non-Fickian Diffusion in a Periodic Potential

Numerical solutions of a non-Fickian diffusion equation belonging to a hyperbolic type are presented in one space dimension.The Brownian particle modelled by this diffusion equation is subjected to a symmetric periodic potential whose spatial shape can be varied by a single parameter.We consider a numerical method which consists of applying Laplace transform in time;we then obtain an elliptic diffusion equation which is discretized using a finite difference method.We analyze some aspects of the convergence of the method.Numerical results for particle density,flux and mean-square-displacement(covering both inertial and diffusive regimes)are presented.

Approximation of the Mean Escape Time for a Tilted Periodic Potential

We present a formula approximating the mean escape time(MST)of a particle from a tilted multi-periodic potential well.The potential function consists of a weighted sum of a finite number of component functions,each of which is periodic.For this particular case,the least period of the potential function is a common period amongst all of its component functions.An approximation of the MST for the potential function is derived,and this approximation takes the form of a product of the MSTs for each of the individual periodic component functions.Our first example illustrates the computational advantages of using the approximation for model validation and parameter tuning in the context of the biological application of DNA transcription.We also use this formula to approximate the MST for an arbitrary tilted periodic potential by the product of MSTs of a finite number of its Fourier modes.Two examples using truncated Fourier series are presented and analyzed.

Exact Electronic Bands for a Periodic Pschl–Teller Potential

We show that supersymmetry is a simple but powerful tool to exactly solve quantum mechanics problems.Here, the supersymmetric approach is used to analyse a quantum system with periodic Pschl–Teller potential, and to find out the exact energy spectra and the corresponding band structure.

Ratchet motion and current motors in pulsating reversal of coupled Brownian symmetric potentials

In this study, we investigate the collective directed transport of coupled Brownian particles in spatially symmetric periodic potentials under time-periodic pulsating modulations. We find that the coupling between two particles can induce symmetry breaking and consequently collective directed motion. Moreover, the direction of motion can be reversed under certain conditions. The dependence of directed current on various parameters is systematically studied, reverse motion can be achieved by modulating the coupling free length and the phase shift of tile pulsating potential. The dynamical mechanism of these transport properties is understood in terms of the effective-potential theory and the space-time transformation invariance. The directed transport of coupled Brownian motors can be maniplflated and optimized by adjusting the coupling strength, pulsating frequency, or noise intensity.

Transition Distributions of Young Diagrams Under Periodically Weighted Plancherel Measures

Kerov[16,17] proved that Wigner＇s semi-circular law in Gauss[an unitary ensembles is the transition distribution of the omega curve discovered by Vershik and Kerov[34] for the limit shape of random partitions under the Plancherel measure. This establishes a close link between random Plancherel partitions and Gauss[an unitary ensembles, In this paper we aim to consider a general problem, namely, to characterize the transition distribution of the limit shape of random Young diagrams under Poissonized Plancherel measures in a periodic potential, which naturally arises in Nekrasov＇s partition functions and is further studied by Nekrasov and Okounkov[25] and Okounkov[28,29]. We also find an associated matrix mode[ for this transition distribution. Our argument is based on a purely geometric analysis on the relation between matrix models and SeibergWitten differentials.

Dirichlet-to-Neumann Mapping for the Characteristic Elliptic Equations with Symmetric Periodic Coefficients

Based on the numerical evidences,an analytical expression of the Dirichletto-Neumann mapping in the form of infinite product was first conjectured for the onedimensional characteristic Schrodinger equation with a sinusoidal potential in[Commun.Comput.Phys.,3(3):641-658,2008].It was later extended for the general secondorder characteristic elliptic equations with symmetric periodic coefficients in[J.Comp.Phys.,227:6877-6894,2008].In this paper,we present a proof for this Dirichlet-toNeumann mapping.

The properties of an asymmetric Gaussian potential quantum well qubit in RbCl crystal

With the circumstance of the electron strongly coupled to LO-phonon and using the variational method of Pekar type（VMPT）,we study the eigenenergies and the eigenfunctions（EE） of the ground and the first excited states（GFES） in a RbCl crystal asymmetric Gaussian potential quantum well（AGPQW）.It concludes：（i） Twoenergy-level of the AGPQW may be seen as a qubit.（ii） When the electron located in the superposition state of the two-energy-level system,the time evolution and the coordinate changes of the electron probability density oscillated periodically in the AGPQW with every certain period T0=22.475 fs.（iii） Due to the confinement that is a two dimensional x-y plane symmetric structure in the AGPQW and the asymmetrical Gaussian potential（AGP） in the AGPQW growth direction,the electron probability density presents only one peak configuration located in the coordinate of z 〉 0,whereas it is zero in the range of z 〈 0.（iv） The oscillatory period is a decreasing function of the AGPQW height and the polaron radius,（v） The oscillating period is a decreasing one in the confinement potential R 〈 0.24 nm,whereas it is an increasing one in the confinement potential R 〉 0.24 nm and it takes a minimum value in R = 0.24 nm.

Two-dimensional matter-wave solitons and vortices in competing cubic-quintic nonlinear lattices

The nonlinear lattice - a new and nonlinear class of periodic potentials - was recently introduced to generate various nonlinear localized modes. Several attempts failed to stabilize two-dimensional （2D） solitons against their intrinsic critical collapse in Kerr media. Here, we provide a possibility for supporting 2D matter-wave solitons and vortices in an extended setting - the cubic and quintic model - by introducing another nonlinear lattice whose period is controllable and can be different from its cubic counterpart, to its quintic nonlinearity, therefore making a fully ＂nonlinear quasi-crystal＂. A variational approximation based on Gaussian ansatz is developed for the fundamental solitons and in particular, their stability exactly follows the inverted Vakhitov-Kolokolov stability criterion, whereas the vortex solitons are only studied by means of numerical methods. Stability regions for two types of localized mode -- the fundamental and vortex solitons -- are provided. A noteworthy feature of the localized solutions is that the vortex solitons are stable only when the period of the quintic nonlinear lattice is the same as the cubic one or when the quintic nonlinearity is constant, while the stable fundamental solitons can be created under looser conditions. Our physical setting （cubic-quintic model） is in the framework of the Gross-Pitaevskii equation or nonlinear Schr6dinger equation, the predicted localized modes thus may be implemented in Bose-Einstein condensates and nonlinear optical media with tunable cubic and quintic nonlinearities.

A Two-Parameter Continuation Method for Rotating Two-Component Bose-Einstein Condensates in Optical Lattices

We study efficient spectral-collocation and continuation methods(SCCM)for rotating two-component Bose-Einstein condensates(BECs)and rotating two-component BECs in optical lattices,where the second kind Chebyshev polynomials are used as the basis functions for the trial function space.A novel two-parameter continuation algorithm is proposed for computing the ground state and first excited state solutions of the governing Gross-Pitaevskii equations(GPEs),where the classical tangent vector is split into two constraint conditions for the bordered linear systems.Numerical results on rotating two-component BECs and rotating two-component BECs in optical lattices are reported.The results on the former are consistent with the published numerical results.