Scaling properties of dynamical localization in monochromatically perturbed quantum maps: Standard map and Anderson map

Dynamical localization phenomena of monochromatically perturbed standard map (SM) and Anderson map (AM), which are both identified with a two-dimensional disordered system under suitable conditions, are investigated by the numerical wave-packet propagation. Some phenomenological formula of the dynamical localization length valid for wide range of control parameters are proposed for both SM and AM. For SM the formula completely agree with the experimental formula, and for AM the presence of a new regime of localization is confirmed. These formula can be derived by the self-consistent mean-field theory of Anderson localization on the basis of a new hypothesis for the cut-off length. Transient diffusion in the large limit of the localization length is also discussed.

Figure 1

The time-dependence of the MSD of SM and AM. (a) Unperturbed SM for $K=3.1,5.1,8.1$ with $\hslash =\frac{2\pi 1248}{2^{15}}$, and for $K=8.1$ with + $\hslash =\frac{2\pi 1248}{2^{14}}$, $\frac{2\pi 1248}{2^{16}}$. (b) Monochromatically perturbed SM for $K=3.1$ and $\hslash =0.12$ with $\epsilon =0.001\sim 0.01$ from below. (c) Unperturbed AM with $W=0.1\sim 0.7$ from top. (d) Monochromatically perturbed AM with some combinations of $\epsilon$ and $W$. Note that the horizontal axes are in logarithmic scale. The system and ensemble sizes are $N=2^{15}\text{–}{2}^{16}$ and $10\text{–}50$, respectively, thorough this paper.

Figure 2

Localization length $p_{\xi }$ as a function of the relatively small perturbation strength $\epsilon$ in the quantum regime ($\epsilon =1\times {10}^{-3}\sim 30\times {10}^{-3}$). (a) $K=3.1$ for $\hslash =\frac{2\pi 435}{2^{12}}$, $\frac{2\pi 435}{2^{13}}$, $\frac{2\pi 435}{2^{14}}$ from below. (b) $K=12.0$ for $\frac{2\pi 1741}{2^{12}}$, $\frac{2\pi 1741}{2^{13}}$, $\frac{2\pi 1741}{2^{14}}$ from below. Note that the horizontal axes are in logarithmic scale.

Figure 3

$(K/\hslash )$ dependencies of the coefficients $A$ and $D$ when varying the combination of $K$ and $\hslash$ variously. Note that the axes are in logarithmic scale. The heavy line shows a straight line with slope of 2 corresponding to the ${(\hslash /K)}^2$.

Figure 4

Localization length of the monochromatically perturbed AM as a function of disorder strength $W$ for various perturbation strength $\epsilon$. The unperturbed case ($\epsilon =0$) is denoted by a thick line with large circles. Note that the axes are in the logarithmic scale.

Figure 5

Localization length of the monochromatically perturbed AM as a function of perturbation strength $\epsilon$ for (a) $W<W^{*}$, (c) $W>W^{*}$, where $W^{*}=0.8$. (b) Plot of $\xi W^2$ as a function of $\epsilon$ for $W<W^{*}$. (d) Plot of $\xi$ as a function of $\epsilon W$ for $W>W^{*}$. Note that the all the vertical axes are in the logarithmic scale. The inset in panel (c) shows the plot of the coefficient $A$ as a function of $W$, which are estimated by linear fitting for data in the panel.

Figure 6

The time dependence of the MSD and the diffusion coefficient $D\left(t\right)$ for relatively large perturbation strength. (a) Monochromatically perturbed SM for $K=3.1$ with $\epsilon =0.016,0.032,0.064,0.09,0.256$ from below. (b) Monochromatically perturbed AM for $W=1.0$ with $\epsilon =0.15,0.2,0.3,0.4,0.5$ from below. (c, d) The $D\left(t\right)$ of the SM and the AM, respectively. Note that the horizontal axes of the $D\left(t\right)$ are in logarithmic scale.

Figure 7

(a) The time-dependent diffusion coefficient $D\left(T\right)$ at $T=1\times {10}^6$ as a function of $\epsilon$ in the monochromatically perturbed AM with $W=1.0,1.2,1.5$. (b) The $D\left(T\right)$ as a function of $W\epsilon$. Note that the horizontal axes are in logarithmic scale.

Figure 8

(a) The time-dependent diffusion coefficient $D\left(T\right)/(K^2/2)$ at $T=4\times {10}^5$ as a function of $\epsilon$ in the monochromatically perturbed SM with $\hslash =\frac{2\pi 1234}{2^{16}}$, $\hslash =\frac{2\pi 1234}{2^{17}}$, $\hslash =\frac{2\pi 1234}{2^{18}}$. Here $K=3.1$ for all data. (b) The $D\left(T\right)$ as a function of $\epsilon /{\hslash }^2$. Note that the horizontal axes are in logarithmic scale.

Figure 9

$K$ dependence of the coefficients (a) $A$ and (b) $D$ of SM with a fixed $\kappa (\equiv K/\hslash )$. The three cases for $\kappa ={\kappa }_0,2{\kappa }_0,4{\kappa }_0$ (${\kappa }_0=18$) are shown from below. The scaled coefficients $A^{*}=A/({\kappa }^2/{\kappa }_{\text{max}})$ and $D^{*}=D/({\kappa }^2/{\kappa }_{\text{max}})$, where ${\kappa }_{\text{max}}=4{\kappa }_0$, are shown for $\kappa ={\kappa }_0,2{\kappa }_0$ and $4{\kappa }_0$ in (c) and (d) of the lower panels, respectively. The three curves almost overlap.

Figure 10

(a) $(\hslash /K)$ dependence of the coefficients (a) $A$ and (b) $D$ for some $K$'s. Three cases of $\kappa ={\kappa }_0,2{\kappa }_0,4{\kappa }_0$ (${\kappa }_0=18$) are shown in the straight lines. Note that the axes are in logarithmic scale. The heavy line shows a straight line with slope of $-2$ corresponding to the ${(\hslash /K)}^{-2}$.

Figure 11

The localization length as a function of scaled perturbation strength $\epsilon W$ in the monochromatically perturbed AM for $\epsilon =0.01,0.02,0.03,0.04$ with perturbation frequencies ${\omega }_1^{\left(2\right)}=\sqrt{2}-1$, ${\omega }_1^{\left(3\right)}=1+1/\sqrt{17}$. All are displayed together.