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Journal of Shaanxi University of Technology

Study of the quantum tunneling from the symmetric to asymmetric double-well potential

Xu Hong

Published Jan 1, 2007• 0 citations
OpenAlex

Abstract

In this paper wave functions of symmetric and asymmetric double-well potential are studied in two-level approximation and it is demonstrated that quantum tunneling varies with the variation of the static electric field strength.The quantum tunneling in the symmetric double well is spontaneous and active,while in asymmetric double well,even the strength of the electric field is little the quantum tunneling disappears completely.Based on the dynamic process of the localized wave function the dynamic vanishing process of quantum tunneling is obtained.

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In this paper wave functions of symmetric and asymmetric double-well potential are studied in two-level approximation and it is demonstrated that quantum tunneling varies with the variation of the static electric field strength.The quantum tunneling in the symmetric double well is spontaneous and active,while in asymmetric double well,even the strength of the electric field is little the quantum tunneling disappears completely.Based on the dynamic process of the localized wave function the dynamic vanishing process of quantum tunneling is obtained.
An asymmetric double-well potential is considered, assuming that the minima of the wells are quadratic with a frequency ω and the difference of the minima is close to a multiple of �ω. A WKB wave function is constructed on both sides of the local maximum between the wells, by matching the WKB function to the exact wave functions near the classical turning points. The continuities of the wave function and its first derivative at the local maximum then give the energy-level splitting formula, which not only reproduces the instanton result for a symmetric potential, but also elucidates the appearance of resonances of tunneling in the asymmetric potential. Key words: quantum tunneling, energy splitting, double-well potential PACS: 03.65.Sq, 03.65.Ge, 82.20.Xr 1
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.
Dynamics of a wave function in a non-symmetrically split (spatially asymmetric) doublewell potential is considered. We study the dependence of the probability of well-to-well transitions on the degree of spatial asymmetry of well sizes and show that the quantum tunneling between the wells is significantly suppressed by this asymmetry. Practically complete suppression occurs at five-ten percent asymmetry. This is close to the threshold of sensitivity of contemporary experimental schemes for creating two-well potentials. We predict the phenomenon of resonance in quantum tunneling of considered states. We have also shown that an incoherently prepared superposition state tunnels in a double-well potential almost in the same way as a perfectly coherent state.
Despite the fact that quantum tunneling has been studied since the advent of quantum mechanics, the literature appears to contain no simple (textbook) formula for tunneling in generic asymmetric double-well potentials. In the regime of strong localization, I derive a succinct analytical formula based on the Wentzel-Kramers-Brillouin semiclassical approach. Two different examples of asymmetric potentials are discussed: the cases when the two localized levels are degenerate and when they are not degenerate. For the first case, I also discuss a time-dependent problem showing the quantum Zeno effect.
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.