Classical Dynamical Localization

We consider classical models of the kicked rotor type, with piecewise linear kicking potentials designed so that momentum changes only by multiples of a given constant. Their dynamics display quasilocalization of momentum, or quadratic growth of energy, depending on the arithmetic nature of the constant. Such purely classical features mimic paradigmatic features of the quantum kicked rotor, notably dynamical localization in momentum, or quantum resonances. We present a heuristic explanation, based on a classical phase space generalization of a well-known argument, that maps the quantum kicked rotor on a tight-binding model with disorder. Such results suggest reconsideration of generally accepted views that dynamical localization and quantum resonances are a pure result of quantum coherence.

Figure 1

The periodic potential $V(\theta )$ (red solid line) is linear in between any two subsequent points ${\theta }_n$ where $\mu \sin ({\theta }_n)/\eta$ takes integer values (only those in $[0,\pi /2]$ are shown). Its piecewise constant slope is $V^{\prime }(\theta )=j(\theta )\eta$ where $j(\theta )$ is the integer that rounds $\mu \sin (\theta )/\eta$ toward 0; here $\mu =3,\eta =1.2$. In the limit of vanishing $\eta$, $V(\theta )$ converges to the standard kicked rotor potential (dashed curve).

Figure 2

$⟨p^2⟩$ of GTMs averaged over an ensemble of ${10}^6$ initial points with ${\theta }_0$ randomly distributed in (0, $2\pi$), versus time $t$ (upper plot) and versus $\log (t{)}^2$ (lower plot). Here $\eta =\pi /\mathrm{G}\mathrm{M}$ (where GM is the golden mean), $\mu =3$, and $p_0=\eta /2,\eta /\sqrt{2},\pi /\sqrt{3}$. Blue lines represent averages over $5\times {10}^6$ initial points (${\theta }_0$, $p_0$) randomly distributed in $(0,2\pi )\times (-\eta /2,\eta /2)$. All curves were averaged over 100 iterations.

Figure 3

Same as Fig. 2 (lower) but with $\mu =4$

Figure 4

Momentum distributions for ensembles of $5\times {10}^6$ initial points (${\theta }_0$, $p_0$) randomly distributed in $(0,2\pi )\times (-\eta /2,\eta /2)$, $\mu =4$, and $\eta /\pi =\mathrm{G}\mathrm{M}$. The curves are averaged over the last 100 kicks.