Delocalization and spreading in a nonlinear Stark ladder

We study the evolution of a wave packet in a nonlinear Stark ladder. In the absence of nonlinearity all normal modes are spatially localized giving rise to an equidistant eigenvalue spectrum and Bloch oscillations. Nonlinearity induces frequency shifts and mode-mode interactions and destroys localization. For large strength of nonlinearity we observe single-site trapping as a transient, with subsequent explosive spreading, followed by subdiffusion. For moderate nonlinearities an immediate subdiffusion takes place. Finally, for small nonlinearities we find linear Stark localization as a transient, with subsequent subdiffusion. For single-mode excitations and weak nonlinearities, stability intervals are predicted and observed upon variation in the dc bias strength, which affects the short- and the long-time dynamics.

Figure 1

(Color online) (a) Black solid line: localization volume $\mathcal{L}$ of the eigenfunction $A_n^{\left(0\right)}$ versus $E$. Insets: explicit form of the eigenfunction across the chain for two values $E=2$ and $E=0.2$ (b, blue; r, red); (b) The diagram of the three regimes of spreading in the parameter space $\left(\beta ,E\right)$. Empty and filled circles: numerically obtained thresholds which separate the three different regimes I–III. Lines connecting symbols guide the eye. Black dashed line: threshold between II and III obtained from the dimer model (see text). In the limit of $E\to \infty$ all lines merge to the asymptotic limit $\beta \propto E$.

Figure 2

(Color online) Single-site excitation for $E=2$. Second moment $m_2$ and participation number $P$ versus time in log-log plots for different values of $\beta$ inside the interval where an explosive delocalization of the trapped regime occurs: $\beta =8.15,8.25,8.5$ (o, orange; g, green; r, red). $\beta =8$ (b, blue): intermediate regime. $\beta =8.9$ (v, violet): trapped regime.

Figure 3

(Color online) ${\left|{\Psi }_n\left(t\right)\right|}^2$ as a function of time (in logarithmic scale) and lattice site for $E=2$, $\beta =8.15$ (upper graph) and $E=0.2$, $\beta =5.2$ (lower graph). The zooms into the explosion region are plotted on a linear time scale. Bloch oscillations are observed right after the explosion time $T_E$.

Figure 4

(Color online) Single-site excitation for $E=2$. $m_2$ and $P$ versus time (log-log plots) for (i) $\beta =2.1$ (b, blue): weak nonlinear regime; (ii) $\beta =4$ (g, green): intermediate regime; and (iii) $\beta =9$ (r, red): trapped regime. Orange dashed line: $m_2\sim t^{0.38}$. Inset: $\zeta$ for $\beta =4$.

Figure 5

(Color online) Amplitudes of the excited, $\nu =0$, and neighboring modes, $\nu =1$ and $\nu =2$, versus time for $\beta =0.4$. (a) $E=1.6$, $\nu =0$ (b, blue) and $E=2$, $\nu =0,1,2$ (o, orange; r, red; g, green). (b) $E=1.18$, $\nu =0$ (b, blue) and $E=1.26$, $\nu =0,1,2$ (o, orange; r, red; g, green). Inset: long time evolution. $m_2$ versus time in log-log plots: $E=1.18,1.26$ (b, blue; o, orange).