Kicked rotator

The Kicked Rotator is a prototype model for chaos and quantum chaos studies. It describes a particle that is free to move on a ring (equivalently: a stick that is free to rotate). The particle is kicked periodically by an homogeneous field (equivalently: the gravitation is switched on periodically in short pulses). The model is described by the Hamiltonian

${\mathcal {H}}(p,x,t)={\frac {1}{2}}p^{2}+K\cos(x)\sum _{n=-\infty }^{\infty }\delta (t-n)$

and its dynamics is described by the standard map. See more details and references there, or better in the associated Scholarpedia entry.

Main properties (classical)

In the classical analysis, if the kicks are strong enough $K>K_{c}\approx 0.971635...$, there is a diffusion in energy space

$\langle (p(t)-p(0))^{2}\rangle =D_{cl}t$

The diffusion coefficient is $D_{cl}\approx K^{2}/2$, because the change $(p(t)-p(0))$ in the momentum is the sum of quasi-random kicks $K\sin(x(n))$. An exact expression for $D_{cl}$ can be obtained in principle by calculating the "area" of the correlation function $C(n)=\langle \sin(x(n))\sin(x(0))\rangle$, namely the sum $D=K^{2}\sum C(n)$. Note that $C(0)=1/2$.

Main properties (quantum)

In the quantum analysis the propagation is realized by iterations with the unitary operator

${\hat {U}}=\exp \left[-i{\frac {1}{2\hbar }}{\hat {p}}^{2}\right]\exp \left[-i{\frac {1}{\hbar }}K\cos {\hat {x}}\right]$

It has been discovered that the classical diffusion is suppressed, and later it has been understood that this is a manifestation of a quantum dynamical localization effect that parallels Anderson Localization.

The effect of noise and dissipation

If noise is added to the system, the dynamical localization is destroyed, and diffusion is induced . This is somewhat similar to hopping conductance. The proper analysis requires to figure out how the dynamical correlations that are responsible for the localization effect are diminished. Without the noise the area under $C(n)$ is zero (due to long negative tails), while with the noise a practical approximation is $C(n)\mapsto C(n)e^{-t/t_{c}}$ where the the coherence time $t_{c}$ is inversely proportional to the intensity of the noise.

Also the problem of quantum kicked rotator with dissipation (due to coupling to a thermal bath) has been considered. There is an issue here how to introduce an interaction that respects the angle periodicity of the position $x$ coordinate, and is still spatially homogeneous. In the first works a quantum-optic type interaction has been assumed that involves a momentum dependent coupling. Later a way to formulate a purely position dependent coupling, as in the Calderia-Leggett model, has been figured out, which can be regarded as the earlier version of the DLD model.

Experiments

Experimental realizations of the quantum kicked rotator have been achieved by the Austin group, and by the Auckland group, and have encouraged a renewed interest in the theoretical analysis: for further reference see .