Fast energy transfer mediated by multi-quanta bound states in a nonlinear quantum lattice

Abstract

By using a Generalized Hubbard model for bosons, the energy transfer in a nonlinear quantum lattice is studied, with special emphasis on the interplay between local and nonlocal nonlinearity. For a strong local nonlinearity, it is shown that the creation of v quanta on one site excites a soliton band formed by bound states involving v quanta trapped on the same site. The energy is first localized on the excited site over a significant timescale and then slowly delocalizes along the lattice. As when increasing the nonlocal nonlinearity, a faster dynamics occurs and the energy propagates more rapidly along the lattice. Nevertheless, the larger is the number of quanta, the slower is the dynamics. However, it is shown that when the nonlocal nonlinearity reaches a critical value, the lattice suddenly supports a very fast energy propagation whose dynamics is almost independent on the number of quanta. The energy is transfered by specific bound states formed by the superimposition of states involving v − p quanta trapped on one site and p quanta trapped on the nearest neighbour sites, with p = 0, .., v − 1. These bound states behave as independent quanta and they exhibit a dynamics which is insensitive to the nonlinearity and controlled by the single quantum hopping constant.

Breathers for the Discrete Nonlinear Schrödinger Equation with Nonlinear Hopping

“…An example is given in [26, Section 2.2 & 2.3, pg. [9][10][11][12][13][14][15][16][17][18], where the minimization of a linear energy functional under a nonlinear constraint verified conditions for the co-existence of breather profiles. For instance, this alternative approach for (2.1) will show the existence of a regime for the hopping parameter α where an upper bound for the power is valid (see Remark II.3).…”

Boundary-induced energy localization in a nonlinear quantum lattice

“…30. This model is based on the numerical observation that, due to the initial creation of v bosons on a single site, the lattice dynamics is confined in a restriction of the v-boson subspace E v .…”

“…According to our previous model, 30 the dynamics in E v is relatively well described by restricting the number state basis to a set of relevant vectors. A relevant vector ͉n , p͘ characterizes v-p quanta of the nth site and p quanta of the site n + 1.…”

“…͑34͒, s =2v⌽ 2 / ͑v −1͒⑀ and ⌽ s stands for the effective hopping constant for v boson bound states as 26,27,30 …”

“…[21][22][23] Within this model, nonlinearity is responsible for a strong interaction between bosons located on the same site. It leads to the occurrence of specific states called multiquantum bound states [14][15][16][17][18][19][20][24][25][26][27][28][29][30][31] which have been observed in various molecular adsorbates [32][33][34][35][36][37][38][39][40] and in ␣-helices. 41,42 A bound state corresponds to the trapping of several quanta over only a few neighboring sites, with a resulting energy which is less than the energy of quanta lying far apart.…”