Decohering localized waves

In the absence of confinement, localization of waves takes place due to randomness or nonlinearity and relies on their phase coherence. We quantitatively probe the sensitivity of localized wave packets to random phase fluctuations and confirm the necessity of phase coherence for localization. Decoherence resulting from a dynamical random environment leads to diffusive spreading and destroys linear and nonlinear localization. We find that maximal spreading is achieved for optimal phase fluctuation characteristics, which is a consequence of the competition between diffusion due to decoherence and ballistic transport within the mean free path distance.

Figure 1

Fully coherent initial conditions that remain localized unless dephasing is applied. (a), (b), (c) NM with eigenenergy close to zero for disordered $(W=5)$, linear growing $(E=5)$, and quasiperiodic $(\zeta =5)$ onsite potentials, respectively. (d) Discrete breather for $(\beta =6)$. The amplitudes are shown by the dots while the black lines guide the eye.

Figure 2

Comparison of (a) $\left|\gamma \right|$ and (b) $m_2$ in the Anderson model after complete random dephasing of a NM. Dephasing is switched off after a number $k$ of kicks: $\left(k\right)=\left(0\right)$ black line, (1) red, (2) green, (3) blue. In (b), only times are shown when the wave packet is, on average, not spreading anymore. A direct correspondence between decoherence and delocalization is observed when changes in the averages of $\left|\gamma \right(\tau \left)\right|$ [dashed lines in (a)] and $m_2\left(t\right)$ [dashed lines in (b)] are compared.

Figure 3

Time averages of $\left|\gamma \right|$ (lower panel), $m_2$ (upper panel) in the (a) Anderson model, (b) Wannier-Stark ladder, (c) Aubry-André model when dephasing is switched off after a small number $k$ of kicks. The averages for $k=1$, 2 and 3 kicks in (a) correspond to the dashed lines in Fig. 2. Up to a number of 8 kicks [in (a), (c), 9 kicks in (b)] an increase (decrease) in $\gamma$ corresponds to a decrease (increase) in $m_2$.

Figure 4

Diffusive spreading in linear chains (Anderson model) for persistent dephasing of initially one normal mode. (a) Complete random dephasing with a time $▵$ between phase kicks of $▵=10$ red (r), 1 green (g), 0.01 blue (b), and 0.001 brown (br). The black (bl) line shows the evolution without dephasing for reference. (b) Quasiperiodic dephasing with strength $b=0.0001$ black (bl), 0.0006 red (r), 0.01 green (g), 0.06 blue (b), 1.1 brown (br).

Figure 5

Diffusive spreading for persistent dephasing of a discrete breather. The crossover from a ballistic to diffusive spreading regime is seen as a jump-like increase of $P$. (a) Complete random dephasing with time $▵$ between phase kicks of $▵=900$ red (r), 500 green (g), 10 blue (b), 0.1 brown (br). The black (bl) line shows the evolution without dephasing for reference. (b) Quasiperiodic dephasing with strength $b=0.0001$ black (bl), 0.000 15 red (r), 0.0002 green (g), 0.000 25 blue (b), 0.0006 brown (br).

Figure 6

Dependence of the diffusion constant $D$ on (a) rate and (b) strength of dephasing in the Anderson model [black circles (bl)], Wannier-Stark ladder [red squares (r)], and Aubry-André model [green diamonds (g)] with the asymptotic dependence in the insets. The occurrence of a global maximum in each model is observed. Each data point of $D$ results from linear regression of an average of $m_2\left(t\right)$ over 100 realisations of sequences of random dephasing. The connecting lines guide the eye.