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arXiv:1209.0752 · Journal of Physics A Mathematical and Theoretical

Tunneling dynamics in exactly solvable models with triple-well potentials

Victor P. Berezovoj, M. I. Konchatnij, A. J. Nurmagambetov

Published Jan 23, 2013• 12 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1751-8113/46/6/065302

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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.
In this paper we propose a self-consistent approach to the description of temporal dynamics of localized states. This approach is based on exactly solvable quantum mechanical models with multi-well potentials and their propagators. States of Hamiltonians with multi-well potentials form a suitable basis for the expansion of wave packets with different shapes and localization degrees. We also consider properties of the tunneling wave packets, taking into account all states of Hamiltonians with symmetric and asymmetric potentials. The study of the dynamics of initially localized states shows that application of the two-state approximation for the description of tunneling is considerably limited, especially for systems which have several states in the under-barrier region, as for example in modern superconducting quantum interference devices and traps for cold atoms.
Abstract We investigate Rosen-Zener tunneling for Bose-Einstein condensates in a triple-well potential within the framework of mean-field treatment. Firstly, we exactly calculate tunneling dynamics for triple-well in the linear case, where the population evolution of each well is robust, and all atoms are finally trapped in the single well, that is initially uploaded. In this case, tunneling dynamics are symmetrical when all atoms are populated in the first well and third well. However, as the nonlinear interaction is introduced, tunneling dynamics will be significantly changed. On the one hand, the symmetry will be broken, some atoms will not be confined to the starting well. On the other hand, nonlinear Josephson oscillation will be presented within a fixing interval. When the interaction exceeds this interval, the self-trapping solution emerges.
We investigate the tunneling dynamics of Bose-Einstein condensates (BECs) in a symmetric as well as in a tilted triple-well trap within the framework of mean-field treatment. The eigenenergies as the functions of the zero-point energy difference between the tilted wells show a striking entangled star structure when the atomic interaction is large. We then achieve insight into the oscillation solutions around the corresponding eigenstates and observe several new types of Josephson oscillations. With increasing the atomic interaction, the Josephson-type oscillation is blocked and the self-trapping solution emerges. The condensates are self-trapped either in one well or in two wells but no scaling law is observed near transition points. In particular, we find that the transition from the Josephson-type oscillation to the self-trapping is accompanied with some irregular regime where tunneling dynamics is dominated by chaos. The above analysis is facilitated with the help of the Poincar\'e section method that visualizes the motions of BECs in a reduced phase plane.
The motion of the center of a wave packet localized in one of the wells of an asymmetric double-well potential is studied. The asymmetric potential is obtained from a solvable symmetric potential using the Gel'fand–Levitan formalism, by keeping the energy eigenvalues unchanged, but changing the normalization of the ground-state wave function. While in a symmetric double well the wave packet tunnels back and forth between the two wells, for an asymmetric well, in general, tunneling does not take place. This is independent of the initial location of the wave packet, i.e., whether it is centered at the minimum of the shallower or of the deeper well. However, when such a system is coupled to a dissipative force, then the tunneling becomes possible. For instance it is shown that if, as a model of dissipative coupling, one chooses Gisin's nonlinear evolution equation, then the center of the wave packet ends up in the deeper well (decay of false vacuum). This result depends on the particular model of dissipation, for instance, an optical potential model yields a different result.
Josephson effect of Bose-Einstein condensate in double-well potential is an obvious manifestation of macroscopic quantum coherence. Most of researches focus on the symmetric double-well potential. In this work, we investigate the dynamic of Bose-Einstein condensates in an asymmetric double-well potential by using two-mode theory and computer simulation. In the absence of the interaction between atoms, the dynamic equation of condensate can be solved analytically. The amplitude as a function of energy difference of two wells is obtained. We can find that the change of energy difference will lead to the different dynamic behaviors of condensate. If the energy difference is relatively large, the condensate will primarily occupy the well that is occupied more than the other well at the beginning time. It is interesting that such a trap phenomenon is not dependent on the position of the high energy potential well nor the position of low energy potential well. If the energy difference becomes small, the tunneling and oscillation of condensate will be enhanced. When the interaction between atoms is present, our numerical calculations show that as the nonlinear interaction increases, the dynamic behavior of condensate exhibits different characteristics, such as trapping in a well, enhancing the tunneling and oscillation between two wells, and enhancing the trapping in a well at large nonlinear interaction, which is similar to the dynamic change caused by the energy difference in the case of ideal condensate. That is to say, on the one hand, the nonlinear interaction can lead to the trap of condensate as well as the dynamic trap to happen in symmetric double-well potential. On the other hand, the nonlinearity can promote the tunneling of condensate, counteracting the effect of the asymmetry of potential. And, this counteracting effect is related to the difference in energy between asymmetric potential wells. To understand the underlying mechanism, the full dynamic behavior of two-mode model is illustrated and the dynamic transition can be seen clearly. Combining the results obtained with and without interaction, regarding nonlinear interaction as effective potential provides a clear way to explain dynamic transition of condensate in an asymmetric double-well potential. In addition, we also perform the numerical simulations of the Gross-Pitaevskii equation, and the results are consistent with the conclusions obtained by using the two-mode theory.