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Dynamics of a Trapped Particle in a Double-Well Potential

Aarouj, Aarouj

Published Jul 1, 2015• 0 citations
OpenAlex

Abstract

Potential wells are widely studied entities nowadays, because of their wide range of applications in many fields of science. Experimental and technological improvements make it possible to design and control the potential shape. For instance, one can modify a single-well potential into a double well and backwards. To investigate this kind of problems, one needs to properly take into account the time-dependency of the system Hamiltonian. We used split operator method to study the dynamics of wavepacket in general, and in time-dependent potential wells in particular. While considering time-dependent potential wells, we concentrated on the case when a single-well is transformed into a double-well. Our goal was to explore the range of validity of the adiabatic theorem. We have made observations on the dynamics of the wavepacket for different barrier heights and widths. For each case, we estimated how long the switching time should be in order to leave the system in the ground state. In doing this, we established a link between the eigenvalues of potential well and the time taken by the wavepacket to be in the ground state of potential well. The greater the difference between the first two eigenvalues of the double well, the lesser the switching time requested to go from single- to double well. Moreover, we simulated the system dynamics while going from single- to asymmetric doublewell. We noticed that the superposition of left and right states is very fragile with respect to asymmetry. By introducing any asymmetry in the double-well, the wavepacket tends to be trapped in the deeper side of the potential well. While talking in terms of eigenvalues of asymmetric double-well, we again observed that the greater the difference between eigenvalues, the lesser the time needed by the wavepacket to be in ground state of double-well. Finally, we applied the theoretically discussed approach to estimate the switching time needed by a trapped ion to be in the ground state of a double-well potential, for realistic experimental parameters.

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Potential wells are widely studied entities nowadays, because of their wide range of applications in many fields of science. Experimental and technological improvements make it possible to design and control the potential shape. For instance, one can modify a single-well potential into a double well and backwards. To investigate this kind of problems, one needs to properly take into account the time-dependency of the system Hamiltonian. We used split operator method to study the dynamics of wavepacket in general, and in time-dependent potential wells in particular. While considering time-dependent potential wells, we concentrated on the case when a single-well is transformed into a double-well. Our goal was to explore the range of validity of the adiabatic theorem. We have made observations on the dynamics of the wavepacket for different barrier heights and widths. For each case, we estimated how long the switching time should be in order to leave the system in the ground state. In doing this, we established a link between the eigenvalues of potential well and the time taken by the wavepacket to be in the ground state of potential well. The greater the difference between the first two eigenvalues of the double well, the lesser the switching time requested to go from single- to double well. Moreover, we simulated the system dynamics while going from single- to asymmetric doublewell. We noticed that the superposition of left and right states is very fragile with respect to asymmetry. By introducing any asymmetry in the double-well, the wavepacket tends to be trapped in the deeper side of the potential well. While talking in terms of eigenvalues of asymmetric double-well, we again observed that the greater the difference between eigenvalues, the lesser the time needed by the wavepacket to be in ground state of double-well. Finally, we applied the theoretically discussed approach to estimate the switching time needed by a trapped ion to be in the ground state of a double-well potential, for realistic experimental parameters.
We consider the evolution of a single-atom wave function in a time-dependent double-well interferometer in the presence of a spatially asymmetric potential. We examine a case where a single trapping potential is split into an asymmetric double well and then recombined again. The interferometer involves a measurement of the first excited state population as a sensitive measure of the asymmetric potential. Based on a two-mode approximation a Bloch vector model provides a simple and satisfactory description of the dynamical evolution. We discuss the roles of adiabaticity and asymmetry in the double-well interferometer. The Bloch model allows us to account for the effects of asymmetry on the excited state population throughout the interferometric process and to choose the appropriate splitting, holding, and recombination periods in order to maximize the output signal. We also compare the outcomes of the Bloch vector model with the results of numerical simulations of the multistate time-dependent Schr\"odinger equation.
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.
We report a rigorous computational treatment of quantum dynamics of cold ions in a double-well trap using the time-dependent Schrödinger equation. Our method employs a numerically accurate approach that avoids approximations, such as assumption of weak coupling between the wells; normal mode nature of vibrations; or harmonic approximation for energy spectrum of the double-well system. Our goal is to reproduce, from first principles, the process of energy swaps between the wells observed in the experiments at NIST [Nature 471, 196 (2011)] and Innsbruck [Nature 471, 200 (2011)]. The model parameters and the initial conditions are carefully chosen to mimic experimental conditions. We obtain accurate energies and wave functions of the system numerically, and study the evolution of motional wave packets to provide new insight. This model reproduces experimental results obtained by NIST and Innsbruck in detail. We explain the energy transfer in terms of wave packet dynamics in the asymmetric potential energy well. We also show that, for a localised initial wave packet, this phenomenon can be interpreted using the terms of classical dynamics, such as trajectory of motion governed by the well-known simple principle: the angle of reflection equals the angle of incidence.
The dynamics of Bose-Einstein condensates in asymmetric double wells is studied. We construct a two-mode model and analyze the properties of the corresponding phase-space diagram, showing in particular that the minimum of the phase-space portrait becomes shifted from the origin as a consequence of the nonvanishing overlap between the ground and excited states from which the localized states are derived. We further incorporate effective interaction corrections in the set of two-mode model parameters. Such a formalism is applied to a recent experimentally explored system, which is confined by a toroidal trap with radial barriers forming an arbitrary angle between them. We confront the model results with Gross-Pitaevskii simulations for various angle values finding a very good agreement. We also analyze the accuracy of a previously employed simple model for moving barriers, exploring a possible improvement that could cover a wider range of trap asymmetries.
The roles of spatial symmetry and strength of external time-dependent perturbation on the dynamics of a quantum particle, initially localized in one of the wells of an asymmetric double-well potential are studied using the recently developed techniques incorporating quantum theory of motion and time-dependent Fourier grid Hamiltonian methods. The model used here includes a mimic of the related experimental situations which is considered as a perturbation to the static double-well potential. Analysis of localized and delocalized phase space structures and corresponding time-profile of tunneling probability reveal the recipe toward controlling the tunneling oscillations by modulating the parameters of applied perturbation. A study on a stochastic pulsating potential also reveals the root to the quantum localization, even in moderate field strength.
We study the dynamics of an interacting classical gas trapped in a double-well potential at finite temperature. Two model potentials are considered: a cubic box with a square barrier in the middle, and a harmonic trap with a Gaussian barrier along one direction. The study is performed using the Boltzmann equation, solved numerically via the test-particle method. We introduce and discuss a simple analytical model that allows one to provide estimates of the relaxation time, which are compared with numerical results. Finally, we use our findings to make numerical and analytical predictions for the case of a fermionic mixture in the normal-fluid phase in a realistic double-well potential relevant for experiments with cold atoms.