Fano Resonances with Discrete Breathers

Abstract

A theoretical study of linear wave scattering by time-periodic spatially localized excitations (discrete breathers (DB)) is presented. We obtain that the wave propagation is strongly influenced by a local coupling between an open and closed channels generated by the DB. A peculiar effect of total reflection occurs due to a Fano resonance when a localized state originating from closed channels resonates with the open channel. For the discrete nonlinear Schrödinger chain we provide with an analytical result for the frequency dependence of the transmission coefficient, including the possibility of resonant reflection. We extend the analysis to chains of weakly coupled anharmonic oscillators and discuss the relevance of the effect for electronic transport spectroscopy of mesoscopic systems. 42.25.Bs, It is a well established fact that various nonlinear spatially discrete systems can support time-periodic spatially localized excitations called discrete breather states (DB) [1]. These states originate from a peculiar interplay between the nonlinearity and discreteness of the lattice rather than from a disorder. While the nonlinearity yields an amplitude-dependent tunability of frequencies of DBs, Ω b , the spatial discreteness of the system leads to finite upper bounds for the frequency spectrum of small amplitude plane waves ω q . This tunability allows one to escape resonances of all multiples of the breather frequency Ω b with the plane wave frequencies ω q , and correspondingly to stabilize the DB state. The frequency dependent localization length of DB's and their stability with respect to small amplitude perturbations have been widely studied [1]. DBs have been observed in experiments covering such diverse fields as interacting Josephson junctions [2], magnetic systems [3] and lattice dynamics of crystals [4].For propagating linear waves a DB acts as a timeperiodic scattering potential, and the transmission coefficient T depends on both the wave vector q of the linear wave and the breather frequency Ω b . The most peculiar effect, observed in many numerical studies of wave scattering by DBs, is the total reflection as T = 0 [5,6]. Note that the presence of a static potential cannot lead to such a total reflection in one-dimensional systems. Similar features are also discussed in other areas, such as electron transport through point contacts, quantum dots and wires [7,8]. The crucial condition allowing a total reflection in these systems is the presence of a few coupled channels connected with the transverse direction of motion. On the other hand, the wave propagation in the presence of a time-periodic scattering potential is characterized by open and closed channels emerging from the Floquet formalism [1,5,6]. The open channel guides the propagating waves, while the eigenfrequencies of closed channels do not match the spectrum of linear waves.In this Letter we show that the total reflection of linear waves in the open channel occurs when a localized state originating from one of the closed channels resonates wi...

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“…(18) and (19) with the boundary conditions 5,6 A(x → +∞) = τ exp(ik x x), A(x → −∞) = exp(ik x x) + ρ exp(−ik x x), and B(x → +∞) = F exp(−κx), B(x → −∞) = D exp(κx). The amplitudes of transmitted, τ , and reflected, ρ, waves in the open channel define the transmission and reflection coefficients T = |τ | 2 = 1 − |ρ| 2 = 1 − R, respectively.…”

“…In general, these harmonics can be either inside or outside the plane wave spectrum, 'open' and 'closed' channels, respectively. 5,6 The presence of closed channels is equivalent to a local increase of the spatial dimensionality, i.e. to an appearance of alternative paths for the plane wave propagation.…”

“…These resonant reflection effects were demonstrated to be similar to the well-known Fano resonance 4 for some nonlinear systems. 5,6 In one-dimensional (1D) systems they can lead to a total resonant reflection of waves. The time-periodic scattering centers can originate from the nonlinearity in a spatially homogeneous system.…”

“…In that limit the resonance position is close to the position of the localized level of the closed channel inside the open channel continuum band (obtained for uncoupled channels). 5,6 Increasing the inter-channel coupling, the resonance position may shift and even completely disappear. We use the weak inter-channel coupling limit as the starting point when searching for the resonance and then follow it while increasing the AC potential strength to its proper value given in Eqs.…”

“…5,6,15 By removing the coupling between the channels, i.e. dropping the term ∼ C 2 , the equation for the A (B) channel becomes equivalent to the stationary Schrödinger equation for an effective particle with the energy…”