Chaos-assisted tunneling in the presence of Anderson localization

Tunneling between two classically disconnected regular regions can be strongly affected by the presence of a chaotic sea in between. This phenomenon, known as chaos-assisted tunneling, gives rise to large fluctuations of the tunneling rate. Here we study chaos-assisted tunneling in the presence of Anderson localization effects in the chaotic sea. Our results show that the standard tunneling rate distribution is strongly modified by localization, going from the Cauchy distribution in the ergodic regime to a log-normal distribution in the strongly localized case, for both a deterministic and a disordered model. We develop a single-parameter scaling description which accurately describes the numerical data. Several possible experimental implementations using cold atoms, photonic lattices, or microwave billiards are discussed.

Figure 1

(a) Splitting distribution $\mathcal{P}(ln\delta )$ for the deterministic model (1) in the localized (left), intermediate (middle), and ergodic (right) regimes. Green dashed line: fit with a log-normal distribution. Blue dashed-dotted line: fit by the Cauchy distribution. Red solid line: analytical theory Eq. (5) with $\mathrm{\Gamma }$ and ${\delta }_{\text{typ}}$ fitting parameters [see Fig. 2]. In the localized (ergodic) regime the log-normal (Cauchy) distributions overlap almost perfectly with Eq. (5). The data correspond to $K=30, p_{\mathrm{r}}=0.4n$ ($n=2048$) and 6400 values of $\hslash$ in a small range around $\hslash =0.85,0.45,0.25$, respectively. (b) Phase-space (Husimi) distribution of the initial wave function (color shades) superimposed on the classical phase space of the deterministic model (1), with $\hslash =0.25, K=30, p_{\text{c}}=10$, and $p_{\mathrm{r}}=1024$. (c) Splittings $\delta$ as a function of $\xi /L$ showing single-parameter scaling behavior. There are 16 sets (different colors) corresponding to fixed values of $K=20,30,40,50$ and $p_{\mathrm{r}}$ varies from $0.1n$ to $0.7n$ ($n=2048$ is the system size) for 1600 values of $\hslash \in [0.1,1.5]$. Big red dots: typical value of $\delta$ averaged over all data sets.

Figure 2

(a) Splitting distribution $\mathcal{P}(ln\delta )$ for the disordered model (3), for $W=3,1,1/2$ from left to right. Note the strong similarity with the distributions for the deterministic model (1) of Fig. 1. Colored lines: as in Fig. 1. $L=256, t_c=0.001, b=3$, and $10000$ realizations (see text). (b) Scaling behavior of ${\delta }_{\text{typ}}$ as a function of $\xi /L$. Disordered model (3): $L=128$ (blue triangles), $L=256$ (magenta squares), and $L=512$ (green stars); parameters as in (a). Black circles: deterministic model (1), $K=30$, 6400 values of $\hslash$ in a small range around $0.25,0.45,0.65,0.85,1.45, p_{\mathrm{r}}=0.4n (n=2048)$. Red solid line: theoretical prediction ${\delta }_{\mathrm{typ}}={\delta }_{\mathrm{ch},\mathrm{typ}}exp(-2L/\xi )$, with ${\delta }_{\mathrm{ch},\mathrm{typ}}$ fitted to $L=512$ data. (c) Scaling behavior of $\mathrm{\Gamma }$ [see Eq. (5)] as a function of $\xi /L$ (symbols defined as in the left panel). Red solid line: theoretical prediction $\mathrm{\Gamma }=L/\xi$.