Delocalization of wave packets in disordered nonlinear chains

We consider the spatiotemporal evolution of a wave packet in disordered nonlinear Schrödinger and anharmonic oscillator chains. In the absence of nonlinearity all eigenstates are spatially localized with an upper bound on the localization length (Anderson localization). Nonlinear terms in the equations of motion destroy the Anderson localization due to nonintegrability and deterministic chaos. At least a finite part of an initially localized wave packet will subdiffusively spread without limits. We analyze the details of this spreading process. We compare the evolution of single-site, single-mode, and general finite-size excitations and study the statistics of detrapping times. We investigate the properties of mode-mode resonances, which are responsible for the incoherent delocalization process.

Figure 1

(Color online) Schematic representations of the three different regimes of spreading for the DNLS (left graph) and the KG model (right graph) in the parameter space of disorder strength $W$ and of the nonlinear frequency shift $\delta$ at initial time $t=0$. For each regime the dependence of $\text{log}{m}_2$ (blue solid curves) and of $\text{log}P$ (red dashed curves) versus $\text{log}t$ are shown schematically (see section III C for details).

Figure 2

(Color online) Single-site excitations. $m_2$ and $P$ versus time in log-log plots. Left plots: DNLS with $W=4$, $\beta =0,0.1,1,4.5$ [(o), orange; (b), blue; (g) green; and (r) red]. Right plots: KG with $W=4$ and initial energy $E=0.05,0.4,1.5$ [(b) blue; (g) green; and (r) red]. The orange (o) curves correspond to the solution of the linear equations of motion, where the term $u_l^3$ in Eq. (7) was absent. The disorder realization is kept unchanged for each of the models. Dashed straight lines guide the eyes for exponents $1/3\left(m_2\right)$ and $1/6\left(P\right)$, respectively. Insets: the compactness index $\zeta$ as a function of time in linear–log plots for $\beta =1$ (DNLS) and $E=0.4$ (KG).

Figure 3

Norm density distributions in the NM space at time $t={10}^8$ for the initial excitations in the regime II of the DNLS model shown in the left plots of Figs. 2, 5. Left plots: single-site excitation for $W=4$ and $\beta =1$. Right plots: single-mode excitation for $W=4$ and $\beta =5$. ${\left|{\varphi }_{\nu }\right|}^2$ is plotted in linear (logarithmic) scale in the upper (lower) plots. The maximal mean value of the localization volume of the NMs $\overline{p}\approx 22$ (shown schematically in the lower plots) is much smaller than the length over which the wave packets have spread.

Figure 4

(Color online) Single-site excitations for the same disorder realization of the KG model. $m_2$ and $P$ versus time in log–log plots. Left panels: plots for $W=4$ and initial energy $E=3.225,4,10$ [(bl) black, (r) red, and (g) green]. Right panels: Plots for $E=0.05$ and $W=6,7$ [(bl) black and (r) red].

Figure 5

(Color online) Single-mode excitations. $m_2$ and $P$ versus time in log–log plots. Left plots: DNLS with $W=4$, $\beta =0,0.6,5,30$ [(o) orange, (b) blue, (g) green, and (r) red]. Right plots: KG with $W=4$ and initial energy $E=0.17,1.1,13.4$ [(b) blue, (g) green, and (r) red]. The orange curves (o) correspond to the solution of the linear equations of motion, where the term $u_l^3$ in Eq. (7) was absent. The disorder realization is kept unchanged for each of the models. Dashed straight lines guide the eye for exponents $1/3\left(m_2\right)$ and $1/6\left(P\right)$, respectively. Insets: the compactness index $\zeta$ as a function of time in linear–log plots for $\beta =5$ (DNLS) and $E=1.1$ (KG).

Figure 6

(Color online) Single-site excitations. $m_2$ (in arbitrary units) versus time in log-log plots in regime II and different values of $W$. Lower set of curves: plain integration (without dephasing); upper set of curves: integration with dephasing of NMs (see Sec. 4A). Dashed straight lines with exponents 1/3 (no dephasing) and 1/2 (dephasing) guide the eye. Left plot: DNLS, $W=4$ and $\beta =3$ (blue); $W=7$ and $\beta =4$ (green); $W=10$ and $\beta =6$ (red). Right plot: KG, $W=10$ and $E=0.25$ (blue); $W=7$ and $E=0.3$ (red); $W=4$ and $E=0.4$ (green). The curves are shifted vertically in order to give maximum overlap within each group.

Figure 7

(Color online) Evolution of $m_2$ versus time in log-log plots. Single-site excitations in the intermediate regime II for the DNLS (a) and the KG model (b) correspond to black curves (bl). The wave packets after $t_d={10}^3$, ${10}^4$, ${10}^5$, and ${10}^6$ time units (t.u.) [(r) red; (g) green; (b) blue; and (p) purple] are registered and relaunched as initial distributions (colored curves). The dashed straight lines correspond to functions $\propto t^{1/3}$.

Figure 8

(Color online) Nonlocal excitations of the KG chain corresponding to initial homogeneous distributions of energy $E=0.4$ over $L$ neighboring sites. (a) $m_2$ and (b) $P$ versus time in log-log plots for $L=1$, 9, 19, 29, and 39 sites [(bl) black; (r) red; (g) green; (b) blue; and (p) purple]. (c) Fitting of the time evolution of $m_2$ for $L=19$ with a curve of the form (18) for $M=3.25$, ${\tau }_d=1052$, and $\alpha =0.303$. (d) The dependence of the detrapping time ${\tau }_d$ on the number $L$ of initially excited sites in log-log scale. The dashed straight line corresponds to a function $\propto L^4$.

Figure 9

(Color online) Statistical properties of NMs of the DNLS model. Probability densities $\mathcal{W}\left(R_{\nu ,{\vec{\mu }}_0}\right)$ of NMs being resonant (see Sec. 4B for details). Disorder strength $W=4,7,10$ (from top to bottom).