Deterministic inhomogeneous inertia ratchets

We study the deterministic dynamics of a periodically driven particle in the underdamped case in a spatially symmetric periodic potential. The system is subjected to a space-dependent friction coefficient, which is similarly periodic as the potential but with a phase difference. We observe that frictional inhomogeneity in a symmetric periodic potential mimics most of the qualitative features of deterministic dynamics in a homogeneous system with an asymmetric periodic potential. We point out the need of averaging over the initial phase of the external drive at small frictional inhomogeneity parameter values or analogously low potential asymmetry regimes in obtaining ratchet current. We also show that at low amplitudes of the drive, where ratchet current is not possible in the deterministic case, noise plays a significant role in realizing ratchet current.     

Research on flux-flow oscillators induced different types of pinning potentials

We have studied dynamics of Josephson vortices in strongly coupled long Josephson junctions stack, such as an intrinsic Josephson junction, by numerical simulations based on coupled sine–Gordon equations considering a periodic pinning potential. In this report, we investigate flux-flow oscillators induced two types of pinning potentials. One is magnetic periodic pinning potential, the other is periodic bias currents. Our results demonstrate that the periodic pinning potential can develop the generated power of flux-flow oscillator in certain condition.     

Controlling the motion of solitons in BEC by a weakly periodic potential

By developing multiple-scale method combined with Wentzel--Kramer--Brillouin expansion, this paper analytically studies the modulating effect of weakly periodic potential on the dynamical properties of the Bose--Einstein condensates (BEC) trapped in harmonic magnetic traps. A black--grey soliton transition is observed in the BEC trapped in harmonic magnetic potential, due to the weakly periodic potential modulating effect. Meanwhile, it finds that with the slight increase of the weakly periodic potential strength, the velocity of the soliton decreases, while its width firstly decreases then increases, a minimum exists there. These results show that the amplitude, velocity, and width of matter solitons can be effectively managed by means of a weakly periodic potential.     

Brownian motors and stochastic resonance

We study the transport properties for a walker on a ratchet potential. The walker consists of two particles coupled by a bistable potential that allow the interchange of the order of the particles while moving through a one-dimensional asymmetric periodic ratchet potential. We consider the stochastic dynamics of the walker on a ratchet with an external periodic forcing, in the overdamped case. The coupling of the two particles corresponds to a single effective particle, describing the internal degree of freedom, in a bistable potential. This double-well potential is subjected to both a periodic forcing and noise and therefore is able to provide a realization of the phenomenon of stochastic resonance. The main result is that there is an optimal amount of noise where the amplitude of the periodic response of the system is maximum, a signal of stochastic resonance, and that precisely for this optimal noise, the average velocity of the walker is maximal, implying a strong link between stochastic resonance and the ratchet effect.     

Effects of periodic optical lattice potential on dark solitons in a Bose Einstein condensate

This paper investigates the dynamics of dark solitons in a Bose--Einstein condensate with a magnetic trap and an optical lattice (OL) trap, and analyses the effects of the periodic OL potential on the dynamics by applying the variational approach based on the renormalized integrals of motion. The results show that the dark soliton becomes only a standing-wave and free propagation of the dark soliton is not possible when the periodic length of the OL potential is approximately equal to the effective width of the dark soliton. When the periodic length is very small or very large, the effects of the OL potential on the dark soliton will be sharply reduced. Finally, the numerical results confirm these theoretical findings.     

Pinning of vortices in a Bose-Einstein condensate by an optical lattice

We consider the ground state of vortices in a Bose-Einstein condensate. We show that turning on a weak optical periodic potential leads to a transition from the triangular Abrikosov vortex lattice to phases where the vortices are pinned by the optical potential. We discuss the phase diagram of the system for a two-dimensional optical periodic potential with one vortex per optical lattice cell. We also discuss the influence of a one-dimensional optical periodic potential on the vortex ground state. The latter situation has no analog in other condensed-matter systems.     

Justification of the Lattice Equation for a Nonlinear Elliptic Problem with a Periodic Potential

We justify the use of the lattice equation (the discrete nonlinear Schrödinger equation) for the tight-binding approximation of stationary localized solutions in the context of a continuous nonlinear elliptic problem with a periodic potential. We rely on properties of the Floquet band-gap spectrum and the Fourier–Bloch decomposition for a linear Schrödinger operator with a periodic potential. Solutions of the nonlinear elliptic problem are represented in terms of Wannier functions and the problem is reduced, using elliptic theory, to a set of nonlinear algebraic equations solvable with the Implicit Function Theorem. Our analysis is developed for a class of piecewise-constant periodic potentials with disjoint spectral bands, which reduce, in a singular limit, to a periodic sequence of infinite walls of a non-zero width. The discrete nonlinear Schrödinger equation is applied to classify localized solutions of the Gross–Pitaevskii equation with a periodic potential.     

周期势场中电子的中心方程及其应用

周期势场中电子的薛定谔微分方程变换为K空间的称为中心方程的线性齐次方程组，利用此方程可以证明布洛赫定理，讨论弱周期势场中电子的能带。     

Bose−Einstein condensates with tunable spin−orbit coupling in the two-dimensional harmonic potential: The ground-state phases,stability phase diagram and collapse dynamics

We study the ground-state phases, the stability phase diagram and collapse dynamics of Bose−Einstein condensates (BECs) with tunable spin−orbit (SO) coupling in the two-dimensional harmonic potential by variational analysis and numerical simulation. Here we propose the theory that the collapse stability and collapse dynamics of BECs in the external trapping potential can be manipulated by the periodic driving of Raman coupling (RC), which can be realized experimentally. Through the high-frequency approximation, an effective time-independent Floquet Hamiltonian with two-body interaction in the harmonic potential is obtained, which results in a tunable SO coupling and a new effective two-body interaction that can be manipulated by the periodic driving strength. Using the variational method, the phase transition boundary and collapse boundary of the system are obtained analytically, where the phase transition between the spin-nonpolarized phase with zero momentum (zero momentum phase) and spin-polarized phase with non-zero momentum (plane wave phase) can be manipulated by the external driving and sensitive to the strong external trapping potential. Particularly, it is revealed that the collapsed BECs can be stabilized by periodic driving of RC, and the mechanism of collapse stability manipulated by periodic driving of RC is clearly revealed. In addition, we find that the collapse velocity and collapse time of the system can be manipulated by periodic driving strength, which also depends on the RC, SO coupling strength and external trapping potential. Finally, the variational approximation is confirmed by numerical simulation of Gross−Pitaevskii equation. Our results show that the periodic driving of RC provides a platform for manipulating the ground-state phases, collapse stability and collapse dynamics of the SO coupled BECs in an external harmonic potential, which can be realized easily in current experiments.     

Localization of a Gaussian polymer in a weak periodic surface potential disturbed by a single defect

Using the results of the recently studied problem of adsorption of a Gaussian polymer in a weak periodic surface potential we study the influence of a single rod like defect on the polymer being localized in the periodic surface potential. We have found that the polymer will be localized at the defect under condition u > u _c, where u_c is the localization threshold in the periodic potential, for any infinitesimal strength of the interaction with defect. We predict that the concentration of monomers of the localized polymer decays exponentially as a function of the distance to the defect and is modulated with the period of the surface potential.     

Resonant tunneling,eigenvalue and energy band calculation for potential and periodic potential structures

Recursion formulae for the reflection and the transmission probability amplitudes and the eigenvalue equation for multistep potential structures are derived. Using the recursion relations, a dispersion equation for periodic potential structures is presented. Some numerical results for the transmission probability of a double barrier structure with scattering centers, the lifetime of the quasi-bound state in a single quantum well with an applied field, and the miniband of a periodic potential structure are presented.     

Calculations of periodic trajectories for the Henon-Heiles Hamiltonian using the monodromy method

The monodromy method, for calculating classical periodic trajectories, is applied to the famous Henon-Heiles potential, which is invariant under the group D(3). The monodromy method is computationally very efficient and is used to find many families of periodic trajectories, including a number of simple bifurcations from the main families of the Henon-Heiles potential.     

Stationary States for Nonlinear Schrödinger Equations with Periodic Potentials

In this paper we consider a one-dimensional non-linear Schrödinger equation with a periodic potential. In the semiclassical limit we prove the existence of stationary solutions by means of the reduction of the non-linear Schrödinger equation to a discrete non-linear Schrödinger equation. In particular, in the limit of large nonlinearity strength the stationary solutions turn out to be localized on a single lattice site of the periodic potential. A connection of these results with the Mott insulator phase for Bose–Einstein condensates in a one-dimensional periodic lattice is also discussed.     

Biased Motion in a Symmetric Periodic Potential by Breaking Temporal Symmetry

Unidirectional transport of a particle in a spatially periodic and symmetric potential under a periodic force with broken temporal symmetry is studied.With a collaboration of the potential field and the asymmetric ac force,a dc current can be observed.Resonant current steps are found for a finite period of the ac force.A phase diagram of these resonant steps is given.Stochastic-resonance-like directional transport induced by thermal nonises is revealed.     

Dynamics of Bose-Einstein condensates in a one-dimensional optical lattice with double-well potential

We study dynamical behaviors of the weakly interacting Bose-Einstein condensate in the one-dimensional optical lattice with an overall double-well potential by solving the time-dependent Gross-Pitaevskii equation. It is observed that the double-well potential dominates the dynamics of such a system even if the lattice depth is several times larger than the height of the double-well potential. This result suggests that the condensate flows without resistance in the periodic lattice just like the case of a single particle moving in periodic potentials. Nevertheless, the effective mass of atoms is increased, which can be experimentally verified since it is connected to the Josephson oscillation frequency. Moreover, the periodic lattice enhances the nonlinearity of the double-well condensate, making the condensate more “self-trapped” in the π-mode self-trapping regime.     

Towards a new proof of Anderson localization

The wave function of a non-relativistic particle in a periodic potential admits oscillatory solutions, the Bloch waves. In the presence of a random noise contribution to the potential the wave function is localized. We outline a new proof of this Anderson localization phenomenon in one spatial dimension, extending the classical result to the case of a periodic background potential. The proof makes use of techniques previously developed to study the effects of noise on reheating in inflationary cosmology, employing methods of random matrix theory.     

A new boundary condition for simulations of systems with charge-charge interactions

We introduce a new boundary condition for simulation of charged systems which is a generalization of periodic boundary conditions and reduces the long-range correlations inherent in periodic boundary conditions. An effective pair potential is calculated for the new condition.     

On the periodic orbits of perturbed Hooke Hamiltonian systems with three degrees of freedom

We study periodic orbits of Hamiltonian differential systems with three degrees of freedom using the averaging theory. We have chosen the classical integrable Hamiltonian system with the Hooke potential and we study periodic orbits which bifurcate from the periodic orbits of the integrable system perturbed with a non-autonomous potential.     

Weak disorder strongly improves the selective enhancement of diffusion in a tilted periodic potential

The diffusion of an overdamped Brownian particle in a tilted periodic potential is known to exhibit a pronounced enhancement over the free thermal diffusion within a small interval of tilt values. Here we show that weak disorder in the form of small, time-independent deviations from a strictly spatially periodic potential may further boost this diffusion peak by orders of magnitude. Our general theoretical predictions are in excellent agreement with experimental observations.     

Interacting Brownian motion in a periodic potential

We study transport properties of a system composed of Brownian particles immersed in a periodic potential. Interaction among the Brownian particles is treated perturbationally in a framework of a generalized Fokker-Planck equation, which due to the interaction contains a renormalized periodic potential and an extra mean field term. We solve the kinetic equation numerically and discuss effects of the (repulsive) interaction on dynamic response functions and transport coefficients.