Tunneling of quantum rotobreathers

Abstract

We analyze the quantum properties of a system consisting of two nonlinearly coupled pendula. This non-integrable system exhibits two different symmetries: a permutational symmetry (permutation of the pendula) and another one related to the reversal of the total momentum of the system. Each of these symmetries is responsible for the existence of two kinds of quasi-degenerated states. At sufficiently high energy, pairs of symmetry-related states glue together to form quadruplets. We show that, starting from the anti-continuous limit, particular quadruplets allow us to construct quantum states whose properties are very similar to those of classical rotobreathers. By diagonalizing numerically the quantum Hamiltonian, we investigate their properties and show that such states are able to store the main part of the total energy on one of the pendula. Contrary to the classical situation, the coupling between pendula necessarily introduces a periodic exchange of energy between them with a frequency which is proportional to the energy splitting between quasi-degenerated states related to the permutation symmetry. This splitting may remain very small as the coupling strength increases and is a decreasing function of the pair energy. The energy may be therefore stored in one pendulum during a time period very long as compared to the inverse of the internal rotobreather frequency.

Discrete breathers — Advances in theory and applications

“…The schematic representation of the appearance of such a quadruplet is shown in Fig. 49 [79]. The obtained quadruplet has an additional fine structure as compared to the tunneling pair of the above considered dimer and trimer.…”

“…Two of them are energy or momentum transfer from one pendulum to the other one, while the third one corresponds to total momentum reversal (which restores time reversal symmetry). The dependence of the corresponding tunneling rates on the coupling ε is shown for a specific quadruplet from [79] in Fig. 50.…”

“…In a recent study Dorignac et al have performed [79] an analysis of the corresponding quantum roto-breather properties in a dimer with the Hamiltonian The classical roto-breather solution consists of one pendulum rotating and the other oscillating with a given period T b . Since the model has two symmetries -permutation of the indices and time-reversal symmetry -which may be both broken by classical trajectories, the irreducible representations of quantum eigenstates contain four symmetry sectors (with possible combinations of symmetric or antisymmetric states with respect to the two symmetry operations).…”

Periodically driven quantum ratchets: Symmetries and resonances

“…It is known that, in the absence of driving, E(t) = 0, the asymptotic dependence of the splitting value on n is △ ∼ n −n , i.e. a superexponential decay [31]. The eigenstates in a plane-wave basis are…”

Dynamical tunnelling in molecules: quantum routes to energy flow

“…Secondly, the various nonlinear resonances are specified and hence it is possible to perform a careful analysis of the role of the resonances and chaos in dynamical tunneling. A third reason has to with the fact that effective Hamiltonians of similar form arise in different contexts like electron transfer through molecular bridges [192], coupled Bose-Einstein condensates, and discrete quantum breathers [193,194,195]. Interestingly the notion of local modes has close connections with the existence of discrete breathers or intrinsic localized modes which have been, and continue to be, studied by a number of researchers [196,198].…”