Time-dependent Potential Well Research Articles

Tunneling Probability of Quantum Wavepacket in Time-Dependent Potential Well

Quantum tunneling of particles plays an important role in many chemical reactions. Studying quantum tunneling in time-dependent potential wells is tricky since most of the available solutions of time-dependent potential wells are coupled to specific properties of the Hamiltonian. Here, we investigate the tunneling probability of a quantum wavepacket in time-dependent potential well by using the split operator method. This numerical method can give us an overview of the tunneling probability of a quantum wavepacket for any temporal change in the shape of the potential well. We study a time-dependent potential well model evolving from symmetric to asymmetric quartic double well since quartic potential wells resemble certain practically available potential well models.Quanta 2024; 13: 11–19.

Exact solutions for time-dependent complex symmetric potential well

Using the pseudo-invariant operator method, we investigate the model of a particle with a time-dependent mass in a complex time-dependent symmetric potential well V (x, t) = if (t) |x|. The problem is exactly solvable and the analytic expressions of the Schrödinger wavefunctions are given in terms of the Airy function. Indeed, with an appropriate choice of the time-dependent metric operators and the unitary transformations, for each region, the two corresponding pseudo-Hermitian invariants transform into a well-known time-independent Hermitian invariant which is the Hamiltonian of a particle confined in a symmetric linear potential well. The eigenfunctions of the last invariant are the Airy functions. Then, the phases obtained are real for both regions and the general solution to the problem is deduced.

Wind-driven transport on the rotating spherical Earth

The dynamics of a water column at the surface of the ocean on the rotating spherical Earth forced by zonal wind stress is analyzed by substituting the angular momentum for the zonal velocity as one of the system's dependent variables. This substitution results in a model of the column's trajectory as a quasiparticle in a time dependent potential well. Explicit solutions are derived for the temporal changes in the angular momentum and the associated minima of the potential well as well as for the oscillations about these minima. The analytic results are confirmed by numerical solutions of the fourth-order nonlinear system of ordinary differential equations. For the eastward directed wind stress, our results provide exact formulas for the time it takes a column to reach the equator, where the dynamics is trivially described by the non-rotating paradigm of a particle subject to a constant force. In mid-latitudes, the analysis underscores the pivotal role played by the latitude where the wind-stress changes sign. Columns originating north or south of this latitude either converge to it or diverge away from it depending on whether the latitudinal change of the wind stress at this latitude is positive or negative. The oscillatory motion about this latitude is linearly unstable, and the growth rate of the amplitude is proportional to the gradient of the wind stress at that latitude.

Multiple Reflections for Classical Particles Moving under the Influence of a Time-Dependent Potential Well

We study the dynamics of classical particles confined in a time-dependent potential well. The dynamics of each particle is described by a two-dimensional nonlinear discrete mapping for the variables energy and phase of the periodic moving well. We obtain the phase space and show that it contains periodic islands, chaotic sea, and invariant spanning curves. We find the elliptic and hyperbolic fixed points and discuss a numerical method to obtain them. We study the dispersion of the initial conditions after a single iteration. This study allows finding regions where multiple reflections occur. Multiple reflections happen when a particle does not have enough energy to exit the potential well and is trapped inside it, suffering several reflections until it has enough energy to exit. We also show deformations in regions with multiple reflection, but the area remains constant when we change the control parameter . Finally, we show some structures that appear in the plane by using density plots.

Chaotic diffusion for particles moving in a time dependent potential well

The chaotic diffusion for particles moving in a time dependent potential well is described by using two different procedures: (i) via direct evolution of the mapping describing the dynamics and; (ii) by the solution of the diffusion equation. The dynamic of the diffusing particles is made by the use of a two dimensional, nonlinear area preserving map for the variables energy and time. The phase space of the system is mixed containing both chaos, periodic regions and invariant spanning curves limiting the diffusion of the chaotic particles. The chaotic evolution for an ensemble of particles is treated as random particles motion and hence described by the diffusion equation. The boundary conditions impose that the particles can not cross the invariant spanning curves, serving as upper boundary for the diffusion, nor the lowest energy domain that is the energy the particles escape from the time moving potential well. The diffusion coefficient is determined via the equation of the mapping while the analytical solution of the diffusion equation gives the probability to find a given particle with a certain energy at a specific time. The momenta of the probability describe qualitatively the behavior of the average energy obtained by numerical simulation, which is investigated either as a function of the time as well as some of the control parameters of the problem.

Scaling and self-similarity for the dynamics of a particle confined to an asymmetric time-dependent potential well

The dynamics of a classical point particle confined to an asymmetric time-dependent potential well is investigated under the framework of scaling. The potential corresponds to a reduced version of a particle moving along an infinitely periodic sequence of synchronously oscillating potential barriers. The dynamics of the model is described by a two-dimensional nonlinear and area preserving map in energy and phase variables. The asymmetric potential well is defined by two regions: Region I with fixed null potential and region II with an oscillating potential. The time-dependent potential of region II makes, for certain initial conditions, the particle to undergo a number of multiple reflections η at the border of the two regions and stay trapped in region I. Such trappings are described by histograms of multiple reflections η, obeying the power-law H(η)∝η^{-ν} with ν≈3, which are scale invariant with a scaling parameter depending of the control parameters of the mapping. We identify the location of the sets of initial conditions in phase space producing the multiple reflections and show that they generate well defined self-similar structures in density plots of trajectories in energy space. The self-similar structures can be enhanced by properly tuning the system parameters.

Fano resonance in the nonadiabatic pumped shot noise of a time-dependent quantum well

We use the Floquet scattering theory to study the correlation properties of the nonadiabatic pumped dc current and heat flow through a time-dependent potential well. Oscillator induced quasibound states of electrons can transit to the Floquet states leading to resonant tunneling effect. Virtual electron scattering processes can produce pumped heat flow, pumped shot noise and pumped heat flow noise, with presence of time and spatial reversal symmetry. When one of the Floquet levels matches the quasibound level there strikes a "Fano" resonance.

Phase space properties and chaotic transport for a particle moving in a time dependent step potential well

Some dynamical properties for an ensemble of non-interacting classical particles along chaotic orbits and transport properties over the chaotic sea for the problem of a step and time dependent potential well are considered. The dynamics of each particle is described by a two-dimensional, nonlinear and area preserving mapping for the variables energy and time. The phase space is of mixed-type and contains periodic islands, a set of invariant KAM curves and chaotic seas. The chaotic orbits are characterized by the use of Lyapunov exponents. Transport over the chaotic sea is considered and scaling exponents are obtained. A sticky region around a chain of periodic islands produces local and temporarily trapping of the dynamics and discussions of the rearrangement of the phase space are made.

Dynamic stabilization of ionization for an atom irradiated by high-frequency laser pulses studied with the Bohmian-trajectory scheme

We study the atomic ionization process in high-frequency laser pulses by the Bohmian-trajectory scheme. Combined with the analysis of the trajectories of Bohmian particles and the time-dependent potential well based on the Kramers-Henneberger transformation, we find that there are two kinds of forces that act on each Bohmian particle (BP): the classical force from the combined potential of the atomic Coulomb potential and the laser field and the quantum force. By investigating the forces acting on different BPs, we find that the quantum force plays an important role in the atomic ionization process in the laser pulse.

Dynamical properties for an ensemble of classical particles moving in a driven potential well with different time perturbation

• We consider dynamical properties for a time-dependent potential well. • The time-dependent potential well is described by different nonlinear functions. • Average properties are characterised as function of the control parameters. We consider dynamical properties for an ensemble of classical particles confined to an infinite box of potential and containing a time-dependent potential well described by different nonlinear functions. For smooth functions, the phase space contains chaotic trajectories, periodic islands and invariant spanning curves preventing the unlimited particle diffusion along the energy axis. Average properties of the chaotic sea are characterised as a function of the control parameters and exponents describing their behaviour show no dependence on the perturbation functions. Given invariant spanning curves are present in the phase space, a sticky region was observed and show to modify locally the diffusion of the particles.

Work distribution in a time-dependent logarithmic–harmonic potential: exact results and asymptotic analysis

We investigate the distribution of work performed on a Brownian particle in a time-dependent asymmetric potential well. The potential has a harmonic component with a time-dependent force constant and a time-independent logarithmic barrier at the origin. For an arbitrary driving protocol, the problem of solving the Fokker–Planck equation for the joint probability density of work and particle position is reduced to the solution of the Riccati differential equation. For a particular choice of the driving protocol, an exact solution of the Riccati equation is presented. An asymptotic analysis of the resulting expression yields the tail behavior of the work distribution for small and large work values. In the limit of a vanishing logarithmic barrier, the work distribution for the breathing parabola model is obtained.

Scaling dynamics for a particle in a time-dependent potential well

Some dynamical properties for a classical particle confined in an infinitely deep box of potential containing a periodically oscillating square well are studied. The dynamics of the system is described by using a two-dimensional non-linear area-preserving map for the variables energy and time. The phase space is mixed and the chaotic sea is described using scaling arguments. Scaling exponents are obtained as a function of all the control parameters, extending the previous results obtained in the literature.

Escape of particles in a time-dependent potential well

We investigate the escape of an ensemble of noninteracting particles inside an infinite potential box that contains a time-dependent potential well. The dynamics of each particle is described by a two-dimensional nonlinear area-preserving mapping for the variables energy and time, leading to a mixed phase space. The chaotic sea in the phase space surrounds periodic islands and is limited by a set of invariant spanning curves. When a hole is introduced in the energy axis, the histogram of frequency for the escape of particles, which we observe to be scaling invariant, grows rapidly until it reaches a maximum and then decreases toward zero at sufficiently long times. A plot of the survival probability of a particle in the dynamics as function of time is observed to be exponential for short times, reaching a crossover time and turning to a slower-decay regime, due to sticky regions observed in the phase space.

Scaling properties for a classical particle in a time-dependent potential well

Some scaling properties for a classical particle interacting with a time-dependent square-well potential are studied. The corresponding dynamics is obtained by use of a two-dimensional nonlinear area-preserving map. We describe dynamics within the chaotic sea by use of a scaling function for the variance of the average energy, thereby demonstrating that the critical exponents are connected by an analytic relationship.

Dynamical properties of a particle in a classical time-dependent potential well

We study numerically the dynamical behavior of a classical particle inside a box potential that contains a square well which depth varies in time. Two cases of time dependence are investigated: periodic and stochastic. The periodic case is similar to the one-dimensional Fermi accelerator model, in the sense that KAM curves like islands surrounded by an ergodic sea are observed for low energy and invariant spanning curves appear for high energies. The ergodic sea, limited by the first spanning curve, is characterized by a positive Lyapunov exponent. This exponent and the position of the lower spanning curve depend sensitively on the control parameter values. In the stochastic case, the particle can reach unbounded kinetic energies. We obtain the average kinetic energy as function of time and of the iteration number. We also show for both cases that the distributions of the time spent by the particle inside the well and the number of successive reflections have a power law tail.

Classical nonlinearity and quantum decay: the effect of classical phase-space structures.

We investigate the decay process from a time dependent potential well in the semiclassical regime. The classical dynamics is chaotic and the decay rate shows an irregular behavior as a function of the system parameters. By studying the weak-chaos regime we are able to connect the decay irregularities to the presence of nonlinear resonances in the classical phase space. A quantitative analytical prediction that accounts for the numerical results is obtained.

The excitation of an independent-particle gas by a time-dependent potential well: Part II

A systematic comparison is carried out between quantal and classical computer simulations of the excitation of independent particles in a time-dependent potential well undergoing one cycle of oscillation. Eleven values of the diffuseness, up to 25 values of the oscillation frequency, and five Legendre polynomial deformations P 2, P 3, P 4, P 5, P 6 were examined. Oscillations around the spherical shape as well as around a P 3 deformed shape were considered. The results are compared with the one-body dissipation theory in the form of the wall formula, including corrections for the heating up of the gas, the diffuseness of the surface, the wave-mechanical suppression of dissipation and the reduction in the effective volume available to a quantal gas in a container whose diffuseness is small compared to the particle wavelengths.

The excitation of an independent-particle gas—classical or quantal—by a time-dependent potential well

Abstract A systematic numerical investigation of the excitation of a classical or quantal gas of non-interacting particles in a time-dependent potential well is described. The excitation energy was followed in time for one oscillation around the sphere for six types of deformation: spheroidal shapes and Legendre polynomial ripples P 2 , P 3 , P 4 , P 5 , P 6 , with relative rms amplitudes of 0.2. Ten different speeds of deformation and eleven different values of the diffuseness of the potential well were studied, making altogether 660 quantal and 660 classical time-dependent calculations. In the upper range of deformation speeds the quantal results for the non-integrable shapes P 3 – P 6 agree approximately with the wall formula for dissipation, the deviations being largely accounted for by the wave-mechanical suppression factor of Koonin et al. For low deformation speeds the dissipation becomes dominated by one or two avoided level crossings.

Noise-induced fluctuations in two-ion systems driven by light beams

The dynamics of a system composed of two strongly coupled ions trapped in a time-dependent potential well and cooled by a laser beam is studied. In the present classical treatment the photon-ion interaction is simulated with an effective noise acting on the ions, which produces a stochastic transfer of momentum. The existence of noise-induced fluctuations in the regime where previous analysis indicated that the two-ion system dynamics was stable is discussed.

A numerical verification of the prediction of an exponential velocity spectrum for a gas of particles in a time-dependent potential well

Using computer simulations we demonstrate that, as predicted by Jarzynski and Swiatecki, the velocity distribution f( ν) for a gas of independent particles bouncing around chaotically in several types of deforming containers tends to an exponential function of ν. The generalized one-body dissipation formula for the heating up of such a gas, also derived by Jarzynski and Swiatecki, is shown to give a good overall account of the numerical results, although small deviations are apparent.