Quantum Perturbation Theory for the Level-Splitting due to Dynamical Tunneling

Vybornyi Evgeny

G. Kalosakas, A. R. Bishop, V. M. Kenkre

The full quantum dynamics of many bosons that are initially completely localized on one site of a symmetric dimer is investigated in the small tunnelling amplitude regime. The number difference of bosons between the two equivalent sites of the dimer exhibits rich behaviour on different timescales, ranging from small amplitude oscillations and collapses and revivals, to coherent tunnelling. We show that this complex quantum evolution is completely accounted for by analytical expressions. A general formula is obtained at a higher order of perturbation theory for the splitting of quasi-degenerate energy levels. The splittings of the two highest pairs provide the characteristic frequencies of the intermediate and long timescale dynamics.

Eli Pollak

The modified version of second and fourth order vibrational perturbation theory, whereby the Euclidean action for tunneling is computed on the inverted potential at a shifted energy that is ℏ2 dependent, is applied to a symmetric double well quartic potential. The mean energies of the doublets in each well are also computed using vibrational perturbation theory. Results show that the modified vibrational perturbation theory significantly improves the estimates of tunneling splitting energies both for the ground state and for excited state doublets.

Ming-Jer Lee, Thomas M. Antonsen, Edward Ott +1 more

Recent numerical experiments of Pecora et al. [Phys. Rev. E 83, 065201 (2011)] have investigated tunneling between two-dimensional symmetric double wells separated by a tunneling barrier. The wells were bounded by hard walls and by the potential barrier which was created by a step increase from the zero potential within a well to a uniform barrier potential within the barrier region, which is a situation potentially realizable in the context of quantum dots. Numerical results for the splitting of energy levels between symmetric and antisymmetric eigenstates were calculated. It was found that the splittings vary erratically from state to state, and the statistics of these variations were studied for different well shapes with the fluctuation levels being much less in chaotic wells than in comparable nonchaotic wells. Here we develop a quantitative theory for the statistics of the energy level splittings for chaotic wells. Our theory is based on the random plane wave hypothesis of Berry. While the fluctuation statistics are very different for chaotic and nonchaotic well dynamics, we show that the mean splittings of differently shaped wells, including integrable and chaotic wells, are the same if their well areas and barrier parameters are the same. We also consider the case of tunneling from a single well into a region with outgoing quantum waves.