Statistical properties of the localization measure in a finite-dimensional model of the quantum kicked rotator

We study the quantum kicked rotator in the classically fully chaotic regime $K=10$ and for various values of the quantum parameter $k$ using Izrailev's $N$-dimensional model for various $N\le 3000$, which in the limit $N\to \infty$ tends to the exact quantized kicked rotator. By numerically calculating the eigenfunctions in the basis of the angular momentum we find that the localization length $\mathcal{L}$ for fixed parameter values has a certain distribution; in fact, its inverse is Gaussian distributed, in analogy and in connection with the distribution of finite time Lyapunov exponents of Hamilton systems. However, unlike the case of the finite time Lyapunov exponents, this distribution is found to be independent of $N$ and thus survives the limit $N=\infty$. This is different from the tight-binding model of Anderson localization. The reason is that the finite bandwidth approximation of the underlying Hamilton dynamical system in the Shepelyansky picture [Phys. Rev. Lett. 56, 677 (1986)] does not apply rigorously. This observation explains the strong fluctuations in the scaling laws of the kicked rotator, such as the entropy localization measure as a function of the scaling parameter $\Lambda =\mathcal{L}/N$, where $\mathcal{L}$ is the theoretical value of the localization length in the semiclassical approximation. These results call for a more refined theory of the localization length in the quantum kicked rotator and in similar Floquet systems, where we must predict not only the mean value of the inverse of the localization length $\mathcal{L}$ but also its (Gaussian) distribution, in particular the variance. In order to complete our studies we numerically analyze the related behavior of finite time Lyapunov exponents in the standard map and of the $2\times 2$ transfer matrix formalism. This paper extends our recent work [Phys. Rev. E 87, 062905 (2013)].

Figure 1

We show $\langle \sigma \rangle$ versus $2/\langle l_H\rangle$ for matrices of dimension $N=3000$, for seven nearby values of $k$, namely, $k=k_0\pm j\delta k$, where $j=0,1,2,3$ and $\delta k=0.00125$, for $k_0=3,4,5,\cdots ,19$. The two empirical localization measures are clearly well defined, linearly related, and thus equivalent.

Figure 2

(a) We show $\mathcal{L}$ versus $2/\langle \sigma \rangle$ for matrices of dimension $N=1000$ (crosses and solid fit line) and for matrices of dimension $N=3000$ (stars and dashed fit line), for seven nearby values of $k$, namely, $k=k_0\pm j\delta k$, where $j=0,1,2,3$ and $\delta k=0.00125$, for $k_0=3,4,5,...,19$. (b) We plot the mean value of $2/(N\langle \sigma \rangle )$ versus ${\beta }_{\mathrm{loc}}$ for $k_0=3,4,5,\cdots ,19$ and seven matrices of dimension $N=3000$ with $k=k_0\pm j\delta k$, where $j=0,1,2,3$ and the step size $\delta k=0.00125$.

Figure 3

We show the histograms of the slopes $\sigma$ for four systems, matrices of dimension $N=3000$, for each of them with seven different values of $k$ close to $k_0=5,9,13,17$, namely, $k=k_0\pm j\delta k$, where $j=0,1,2,3$ and $\delta k=0.00125$: (a) $k_0=5$, (b) $k_0=9$, (c) $k_0=13$, and (d) $k_0=17$.

Figure 4

We show the histograms of $l_H$ in (a) and $2/{ł}_H$ in (b) for the system $k=10$ described by the matrices of dimension $N=3000$. In both cases we show the Gaussian best fit.

Figure 5

We show log-log plots in (a) the mean slope $\langle \sigma \rangle$ as a function of $k$ and in (b) the standard deviation of $\sigma$ as a function of $k$. The fitting by a straight line is only on the semiclassical interval $10\le k\le 19$. In the former case the behavior is roughly as $1/k^2$, in agreement with the theoretical estimate $1/k^2$ of Eq. (12), and in the latter case also like $1/k^2$, surely not as the theoretical estimate $1/k$ based on the Lyapunov exponents method in Ref. [41] [Eq. (9)].

Figure 6

We show the histograms of the positive finite time Lyapunov exponents for the standard map [Eq. (3)] with $K=10$ and times (number of iterations) $t=50,100,500,1000$ in (a), (b), (c), and (d), respectively. The initial conditions are on the grid $200\times 200$ on the square $[0,2\pi )\times [0,2\pi )$. In all cases we show the Gaussian best fit.

Figure 7

The histograms of the positive finite time Lyapunov exponents for the product of random matrices [Eq. (26)] with $W=E-E_n^0$ uniformly distributed in a box $W\in [-2,+2]$, for $n=50,100,500,1000$ in (a), (b), (c), and (d), respectively. In all cases we show the Gaussian best fit.

Figure 8

The histograms of the positive finite time Lyapunov exponents for the product of random matrices [Eq. (26)] at $n=100$ with (a) $W=E-E_n^0$ Gaussian distributed with zero mean and unit variance, (b) Cauchy-Lorentz distribution [Eq. (27)] with $W$ in the cutoff interval $[-2,+2]$, and (c) the same as (b) but $W\in [-100,+100]$. In all cases we show the Gaussian best fit, which is excellent.

Figure 9

The standard deviation of the positive finite time Lyapunov exponents for the standard map (stars) and for the product of random transfer matrices with the box distribution of $W$ (empty boxes), as a function of time in $log-log$ presentation, and their best fits. The slope is exactly $-1/2$.