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International Journal of Quantum Chemistry

Tunneling and quantum localization in chaos‐driven symmetric triple well potential: An approach using quantum theory of motion

Susmita Kar, Pratim Kumar Chattaraj

Published Nov 3, 2017• 5 citations
Publisher

Abstract

Abstract For a symmetric triple well potential, driven by the forces associated with the bifurcation diagram of a logistic map, the tunneling and quantum localization are studied using quantum theory of motion and time‐dependent Fourier grid Hamiltonian methods. Detailed analysis reveals that application of only asymmetric or symmetric perturbation results into either quantum localization or over‐barrier transition and no tunneling while application of mixed symmetry perturbation gives either tunneling or over‐barrier transition, depending on temporal nature and initial position of the particle. For bifurcative and chaotic symmetric‐asymmetric perturbation, with truncation of mixed symmetry perturbation, a sudden jump in energy causes a transition from the tunneling phenomenon to the over‐barrier transition. With particle located initially near to either of the minima of the unperturbed well, quantum localization, or over‐barrier transition is observed, depending on types of perturbation used.

DOI
10.1002/qua.25531

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Abstract For a symmetric triple well potential, driven by the forces associated with the bifurcation diagram of a logistic map, the tunneling and quantum localization are studied using quantum theory of motion and time‐dependent Fourier grid Hamiltonian methods. Detailed analysis reveals that application of only asymmetric or symmetric perturbation results into either quantum localization or over‐barrier transition and no tunneling while application of mixed symmetry perturbation gives either tunneling or over‐barrier transition, depending on temporal nature and initial position of the particle. For bifurcative and chaotic symmetric‐asymmetric perturbation, with truncation of mixed symmetry perturbation, a sudden jump in energy causes a transition from the tunneling phenomenon to the over‐barrier transition. With particle located initially near to either of the minima of the unperturbed well, quantum localization, or over‐barrier transition is observed, depending on types of perturbation used.
Abstract The effects of discontinuously time‐varying perturbations on the dynamics of a particle moving in harmonic, symmetric double well and symmetric triple well potentials, are investigated both classically and quantum mechanically. The quantum dynamics is followed using the time‐dependent Fourier grid Hamiltonian (TDFGH) method while the classical dynamics is analyzed within the framework of classical Hamiltonian mechanics. Depending on the spatial symmetry of the perturbation and the characteristic features of the reversal time , different types of ‘phase space’ structures are observed in each of the potentials. For symmetric double and triple well potentials, quantum dynamics reveals that complete destruction of tunnelling (CDT) can be achieved in the presence of a time‐dependent spatially asymmetric perturbing field that is continuous in time. Any discontinuity in time‐variation of the perturbation may induce over the barrier transition. The relevance of these results in the context of (i) tunnelling control and (ii) quantum computing with 3‐state or 2‐state quantum registers is briefly discussed.
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.
The behavior of a driven symmetric triple well potential has been studied by developing an algorithm where the well‐established Bohmian mechanics and time‐dependent Fourier Grid Hamiltonian method are incorporated and the quantum theory of motion (QTM) phase space structures of the particle are constructed, both in “nonclassical” and “classical” limits. Comparison of QTM phase space structures with their classical analogues shows both similarity as well as dissimilarities. The temporal nature and the spatial symmetry of applied perturbation play crucial roles in having similar phase space structures. © 2016 Wiley Periodicals, Inc.
Abstract We calculate quantum tunneling time in the double‐well system without perturbation and with symmetry‐breaking driven force using the entangled trajectory molecular dynamics method in the present article. Without perturbation, quantum tunneling time decreased with the increase of the energy, and the different contributions of the barrier traversal time and the intrinsic decay time have been shown. The tunneling time dependence on the amplitude and frequency of the symmetry‐breaking driven force are present. In the case of weak driven force, tunneling time has a minimum value in the resonant frequency. For strong driven force, chaos brings a huge change to quantum dynamics, tunneling time significantly becomes short. Finally, we directly show the enhancement of quantum tunneling process by chaotic behavior of entangled trajectories and indicate that it was caused by quantum effect.
Inspired by new trends in atomtronics, cold atom devices and Bose–Einstein condensate dynamics, we apply a general technique of N = 4 extended supersymmetric quantum mechanics to isospectral Hamiltonians with triple-well potentials, i.e. symmetric and asymmetric. Expressions of quantum-mechanical propagators, which take into account all states of the spectrum, are obtained, within the N = 4 SQM approach, in the closed form. For the initial Hamiltonian of a harmonic oscillator, we obtain the explicit expressions of potentials, wavefunctions and propagators. The obtained results are applied to tunneling dynamics of localized states in triple-well potentials and for studying its features. In particular, we observe a Josephson-type tunneling transition of a wave packet, the effect of its partial trapping and a non-monotonic dependence of tunneling dynamics on the shape of a three-well potential. We investigate, among others, the possibility of controlling tunneling transport by changing parameters of the central well, and we briefly discuss potential applications of this aspect to atomtronic devices.
Since the early days of quantum mechanics, tunneling has been recognized as one of the hallmarks of the wave character of microscopic physics. The possibility of a quantum particle to penetrate an energetic barrier represents certainly one of the most spectacular implications of quantum theory and has led to various applications in nuclear, atomic, and molecular physics as well as in mesoscopic science. Typical scenarios in which tunneling manifests are the escape route of a quantum particle from a quasibounded region, the transition between two or more symmetry-related, but classically disconnected wells (which we shall focus on in the following), as well as scattering or transport through potential barriers. The spectrum of scenarios becomes even richer when the concept of tunneling is generalized to any kind of classically forbidden transitions in phase space, that is, to transitions that are not necessarily inhibited by static potential barriers but by some other constraints of the underlying classical dynamics (such as integrals of motion). Such “dynamical tunneling” processes arise frequently in molecular systems [1] and were realized with cold atoms propagating in periodically modulated optical lattices [2-4]. Moreover, the electromagnetic analog of dynamical tunneling was also obtained with microwaves in billiards [5].