Sinusoidal Potentials Research Articles

Langevin’s Equation of Motion: Dynamical States and Stochastic Resonance

In this present work, we numerically observe the existence of two dynamical states of trajectories in the underdamped motion of a particle in an inhomogeneous periodic bistable potential with high friction coefficient driven by a periodic force. The system is inhomogeneous in the sense that the coefficient of friction is periodic in space and with phase difference 'ϕ' with respect to the periodic bistable potential. Just like in sinusoidal potentials, the occurrence of stochastic resonance (SR) in the periodic bistable potential is due to the transitions between the dynamical states at an optimal (thermal) noise strength.

Topological charge pumping with subwavelength Raman lattices

Recent experiments demonstrated deeply subwavelength lattices using atoms with $N$ internal states Raman coupled with lasers of wavelength $\ensuremath{\lambda}$. The resulting unit cell was $\ensuremath{\lambda}/2N$ in extent, an $N$-fold reduction compared to the usual $\ensuremath{\lambda}/2$ periodicity of an optical lattice. For resonant Raman coupling, this lattice consists of $N$ independent sinusoidal potentials (with period $\ensuremath{\lambda}/2$) displaced by $\ensuremath{\lambda}/2N$ from each other. We show that detuning from Raman resonance induces tunneling between these potentials. Temporally modulating the detuning couples the $s$ and $p$ bands of the potentials, creating a pair of coupled subwavelength Rice-Mele chains. This operates as a topological charge pump that counterintuitively can give half the displacement per pump cycle of each individual Rice-Mele chain separately. We analytically describe this behavior in terms of infinite-system Chern numbers and numerically identify the associated finite-system edge states.

Modulating drift dynamics of circle swimmers by periodic potentials.

We propose a method to modulate the drifting motion of overdamped circle swimmers in steady fluid flows by means of static sinusoidal potentials. Using Langevin formalism, we study drift velocity as a function of potential strength and wavelength with and without diffusional motion. Drift velocity is essentially quantized without diffusion, but in the presence of noise, the displacement per cycle has a continuous range. As a function of dimensionless potential wave number, domains of damped oscillatory and plateau regimes are observed in the drift velocity diagram. At weak potential and fluid velocity less than powered velocity, there is also a regime where drift velocity exceeds the fluid velocity. Methods based on these results can be used to separate biological and artificial circle swimmers based on their dynamical properties.

Spectral Asymptotics and Lamé Spectrum for Coupled Particles in Periodic Potentials

We make two observations on the motion of coupled particles in a periodic potential. Coupled pendula, or the space-discretized sine-Gordon equation is an example of this problem. Linearized spectrum of the synchronous motion turns out to have a hidden asymptotic periodicity in its dependence on the energy; this is the gist of the first observation. Our second observation is the discovery of a special property of the purely sinusoidal potentials: the linearization around the synchronous solution is equivalent to the classical Lamè equation. As a consequence, all but one instability zones of the linearized equation collapse to a point for the one-harmonic potentials. This provides a new example where Lamé’s finite zone potential arises in the simplest possible setting.

Floquet–Bloch eigenwaves and bandgaps in a di-periodic potential

We present general solutions to the Schrödinger equation for a di-periodic potential composed of two frequencies, thus generalizing the standard sinusoidal potential. The Schrödinger equation with a di-periodic potential becomes a three-term Whittaker–Hill equation that is solved by two different approaches. The first is applying the central equation formalism, which allows determining the Floquet–Bloch eigenwaves and band structure as a function of the potential parameters. In the second approach, we transform the Whittaker–Hill equation into an Ince equation with a suitable change of variable. In this case, we get a complete set of orthogonal solutions described by Ince functions. The well-known Mathieu functions obtained with purely sinusoidal potentials are a special case of Ince functions.

Online and offline effects of transcranial alternating current stimulation of the primary motor cortex

Transcranial alternating current stimulation (tACS) is a non-invasive brain stimulation technique that allows interaction with endogenous cortical oscillatory rhythms by means of external sinusoidal potentials. The physiological mechanisms underlying tACS effects are still under debate. Whereas online (e.g., ongoing) tACS over the motor cortex induces robust state-, phase- and frequency-dependent effects on cortical excitability, the offline effects (i.e. after-effects) of tACS are less clear. Here, we explored online and offline effects of tACS in two single-blind, sham-controlled experiments. In both experiments we used neuronavigated transcranial magnetic stimulation (TMS) of the primary motor cortex (M1) as a probe to index changes of cortical excitability and delivered M1 tACS at 10 Hz (alpha), 20 Hz (beta) and sham (30 s of low-frequency transcranial random noise stimulation; tRNS). Corticospinal excitability was measured by single pulse TMS-induced motor evoked potentials (MEPs). tACS was delivered online in Experiment 1 and offline in Experiment 2. In Experiment 1, the increase of MEPs size was maximal with the 20 Hz stimulation, however in Experiment 2 neither the 10 Hz nor the 20 Hz stimulation induced tACS offline effects. These findings support the idea that tACS affects cortical excitability only during online application, at least when delivered on the scalp overlying M1, thereby contributing to the development of effective protocols that can be applied to clinical populations.

Stochastic resonance and free oscillation in a sinusoidal potentials driven by a square-wave periodic force

Recently, the occurrence of stochastic resonance in a sinusoidal potential driven by a sinusoidal force and a Gaussian white noise was experimentally verified. In this work, we experimentally show that stochastic resonance in sinusoidal potentials can also be observed when driven by a square-wave periodic force. The occurrence of stochastic resonance could be explained as due to the appearance of two dynamical states similar to what was done when driven by a sinusoidal force at large frequencies. However, at smaller frequencies of the square-wave drive, the free but damped oscillations of the output could be observed and the effective exponential damping coefficients measured.

A hyper-block self-consistent approach to nonlinear Schrodinger equations: Breeding, metamorphosis, and killing of Hofstadter Butterflies

Nonlinear Schrödinger equations play essential roles in different physics and engineering fields. In this paper, a hyper-block finite-difference self-consistent method (HFDSCF) is employed to solve this stationary nonlinear eigenvalue equation and demonstrated its accuracy. By comparing the results with the Sinc self-consistent (SSCF) method and the exact available results, we show that the HFDSCF gives quantum states with high accuracy and can even solve the strongly nonlinear Schrodinger equations. Then, by applying our method to the Hofstadter butterfly problem, we describe the breeding, metamorphosis, and killing of these butterflies by using nonlinear interactions and two constant length multi-well and sinusoidal potentials.

Kinetic friction force in a three-layer model with commensurate substrates

A chain of harmonically interacting particles confined between two sinusoidal substrate potentials has been analysed. When the top substrate is driven through an attached spring with a constant velocity, the influence of the system parameters on the kinetic friction force was examined in the system with commensurate substrates. It was shown that the friction reached its maximum value when the equilibrium distance of the intermediate particles was an integer multiple of the space period of the upper and lower substrate potentials, which led to the maximum energy loss. This loss can be reduced by the increase of interparticle interaction of the middle layer.

Dependence of static friction force on stiffness in a confined chain with the incommensurate and commensurate structrue

It is used that a series of interacting atoms. They are confined between two sinusoidal substrate potentials. When the top substrate is driven by the external driving force, its velocity changes from zero to non-zero. The dependence of the static friction force on the strength is presented for the incommensurate(the spiral mean and the golden mean ) and commensurate structrue. It is found that incommensurate structures is more likely to produce superlubricate phenomenon more than commensurate structures under different stiffness coefficients.

Electro-osmotic oscillatory flow of viscoelastic fluids in a microchannel

This work presents an analytical solution for electro-osmotic flow (EOF) in small amplitude oscillatory shear (SAOS) as a measuring tool suitable to characterize the linear viscoelastic properties of non-Newtonian fluids in microchannel flow. The flow in the straight microchannel is driven by applying oscillating sinusoidal electric potentials. Fourier series are used to derive an expression for the velocity field, under an externally imposed generic potential field aimed at the practical application of SAOS in characterizing the rheological properties of viscoelastic fluids. This extensive investigation covers a wide range of parameters and considers the multi-mode upper-convected Maxwell (UCM) model, which represents the rheology of viscoelastic fluids in the limit of small and slow deformations. Particular focus is given to two cases of practical interest: equal wall zeta potentials at both channel walls and negligible zeta potential at one of the walls. The results show that in flows with thin electric double layers (EDL), Reynolds numbers below 0.001, and Deborah numbers below 100, corresponding to a viscoelastic Mach number of 0.32, the velocity field outside the EDLs is linear and has a large enough amplitude of oscillation, which may allow the quantification of the storage and loss moduli in the linear regime. This technique requires the use of significantly smaller sample sizes than the traditional SAOS in a rotational rheometer.

Realizing the Harper model with ultracold atoms in a ring lattice

We demonstrate that all of the salient features of the Harper-Hofstadter model can be implemented with ultracold atoms trapped in a bichromatic ring-shaped lattice. Using realistic sinusoidal lattice potentials rather than assume the idealized tight-binding picture, we determine the optimal conditions necessary to realize the critical point where the spectrum becomes fractal, and identify the nature and cause of the departures from the discrete model predictions. We also show that even with a commensurate ring with a few lattice sites, the Aubry-Andr\'e localization transition can be realized. Localized states that behave like edge states with energies that reside in the band gaps can be generated by introducing a surprisingly small local perturbation within the ring. Spectrum oscillation arising from complex coupling can be implemented by uniform rotation of the ring, but with certain significant differences that are explained

Resonance oscillation of a damped driven simple pendulum

The resonance characteristics of a driven damped harmonic oscillator are well known. Unlike harmonic oscillators which are guided by parabolic potentials, a simple pendulum oscillates under sinusoidal potentials. The problem of an undamped pendulum has been investigated to a great extent. However, the resonance characteristics of a driven damped pendulum have not been reported so far due to the difficulty in solving the problem analytically. In the present work we report the resonance characteristics of a driven damped pendulum calculated numerically. The results are compared with the resonance characteristics of a damped driven harmonic oscillator and also with those of a damped driven Duffing oscillator. The work can be of pedagogic interest too as it reveals the richness of driven damped motion of a simple pendulum in comparison to, and how strikingly it differs from, the motion of a driven damped harmonic oscillator. We confine our work only to the nonchaotic regime of pendulum motion.

Stochastic resonance in underdamped periodic potential systems with alpha stable Lévy noise

In this paper, we investigate the effect of alpha stable Lévy noise with alpha stability index α (0<α≤2) on stochastic resonance (SR) in underdamped periodic potential systems by the non-perturbative expansion moment method and stochastic simulation. Using the spectral amplification factor as a quantifying index, we find that SR can occur in both sinusoidal potentials and ratchet potentials when α is close to 2, while the resonant effect becomes weaker as the stability index decreases. By means of massive numerical statistics, we ascribe this trend to the typical jumps of non-Gaussian Lévy noise (0<α<2), which play a destructive role on the periodicity of the long time mean response. We also disclose that the skewness parameter of Lévy noise has a more notable impact on the resonant effect of the asymmetric ratchet potential than that of the symmetric sinusoidal potential because of symmetry breaking.

Implementation strategies for multiband quantum simulators of real materials

The majority of quantum simulators treat simplified one-band strongly correlated models, whereas multiple bands are needed to describe materials with intermediate correlation. We investigate the sensitivity of multiband quantum simulators to: (1) the form of optical lattices (2) the interactions between electron analogues. Since the kinetic energy terms of electron analogues in a quantum simulator and electrons in a solid are identical, by examining both periodic potential and interaction we explore the full problem of many-band quantum simulators within the Born-Oppenheimer approximation. Density functional calculations show that bandstructure is highly sensitive to the form of optical lattice, and it is necessary to go beyond sinusoidal potentials to ensure that the bands closest to the Fermi surface are similar to those in real materials. Analysis of several electron analogue types finds that dressed Rydberg atoms (DRAs) have promising interactions for multi band quantum simulation. DRA properties can be chosen so that interaction matrices approximate those in real systems and decoherence effects are controlled, albeit with parameters at the edge of currently available technology. We conclude that multiband quantum simulators implemented using the principles established here could provide insight into the complex processes in real materials.

Dependence of static friction force on stiffness and lubricant structure in a confined chain

We use a chain of harmonically interacting atoms confined between two sinusoidal substrate potentials in which the top substrate is driven by the driving force. This system is closely related to physical situations with confined lubricant films. The dependence of the static friction force on the system parameters such as the equilibrium spacing and the strength is presented. It is found that the better lubricant films are materials with larger interatomic interaction strength, and the lubricant structure is in the certain region.

Electrophysiological representation of scratching CpG activity in the cerebellum.

We analyzed the electrical activity of neuronal populations in the cerebellum and the lumbar spinal cord during fictive scratching in adult decerebrate cats before and after selective sections of the Spino-Reticulo Cerebellar Pathway (SRCP) and the Ventral-Spino Cerebellar Tract (VSCT). During fictive scratching, we found a conspicuous sinusoidal electrical activity, called Sinusoidal Cerebellar Potentials (SCPs), in the cerebellar vermis, which exhibited smaller amplitude in the paravermal and hemisphere cortices. There was also a significant spino-cerebellar coherence between these SCPs and the lumbar sinusoidal cord dorsum potentials (SCDPs). However, during spontaneous activity such spino-cerebellar coherence between spontaneous potentials recorded in the same regions decreased. We found that the section of the SRCP and the VSCT did not abolish the amplitude of the SCPs, suggesting that there are additional pathways conveying information from the spinal CPG to the cerebellum. This is the first evidence that the sinusoidal activity associated to the spinal CPG circuitry for scratching has a broad representation in the cerebellum beyond the sensory representation from hindlimbs previously described. Furthermore, the SCPs represent the global electrical activity of the spinal CPG for scratching in the cerebellar cortex.

Multi-natural inflation in supergravity

We show that the recently proposed multi-natural inflation can be realized within the framework of 4D ${\cal N}=1$ supergravity. The inflaton potential mainly consists of two sinusoidal potentials that are comparable in size, but have different periodicity with a possible non-zero relative phase. For a sub-Planckian decay constant, the multi-natural inflation model is reduced to axion hilltop inflation. We show that, taking into account the effect of the relative phase, the spectral index can be increased to give a better fit to the Planck results, with respect to the hilltop quartic inflation. We also consider a possible UV completion based on a string-inspired model. Interestingly, the Hubble parameter during inflation is necessarily smaller than the gravitino mass, avoiding possible moduli destabilization. Reheating processes as well as non-thermal leptogenesis are also discussed.

Multi-natural inflation

We propose a multi-natural inflation model in which the single-field inflaton potential consists of two or more sinusoidal potentials that are comparable in size but have different periodicity with a possible non-zero relative phase. The model is versatile enough to realize both large-field and small-field inflation. We show that, in a model with two sinusoidal potentials, the predicted values of the spectral index and tensor-to-scalar ratio lie within the 1σ region of the Planck data. In particular, there is no lower bound on the decay constants in contrast to the original natural inflation. We also show that, in a certain limit, multi-natural inflation can be approximated by a hilltop quartic inflation model.

Stimulated neutrino transformation with sinusoidal density profiles

Large amplitude oscillations between the states of a quantum system can be stimulated by sinusoidal external potentials with frequencies that are similar to the energy level splitting of the states or a fraction thereof. Situations where the applied frequency is equal to an integer fraction of the energy level splittings are known as parametric resonances. We investigate this effect for neutrinos both analytically and numerically for the case of arbitrary numbers of neutrino flavors. We look for environments where the effect may be observed and find that supernovae are the one realistic possibility due to the necessity of both large densities and large amplitude fluctuations. The comparison of numerical and analytical results of neutrino propagation through a model supernova reveals that it is possible to predict the locations and strengths of the stimulated transitions that occur.