D.R. Grempel, R.E. Prange and S. Fishman, Phys. Rev. A, 29, 1639 (1984);

Figure 1: Two quasi-energy eigenstates obtained numerically. Quasi-energies are ω =2πj/210 with j =323 (solid circles) and j =621 (open circles). The potential is V(θ)=-2arctan(cosθ-E).

Figure 2: Two quasi-energy eigenstates obtained numerically. Quasi-energies are ω =2πj/210 with j =511 (solid circles) and j =709 (open circles). The potential V(θ)=kcosθ.

D.R. Grempel, R.E. Prange and S. Fishman, Phys. Rev. A, 29, 1639 (1984);

Figure 3: The time dependence of the kinetic energy of the quantum kicked rotor for k=20 and K=4. The motion is reversed at t=150 (vertical line) and small noise is added. The quantum evolution is completely reversible. The straight line corresponds to classical diffusion.

t

D.L. Shepelyansky, Physica D, 8, 208 (1983).

Figure 4: The ratio between ξ, the localization length of the kicked rotor with =1 an =2, and ξR, the localization length of the corresponding random model.

R. Blumel, S. Fishman, M. Griniasty and U. Smilansky, in Quantum Chaos and Statistical Nuclear Physics, Proc. of the 2nd International Conference on Quantum Chaos, Curnevaca, Mexico, Edited by T.H. Seligman and H. Nishioka, (Springer-Verlag, Heidelberg, 1986).}

All the foregoing figures also appear at: S. Fishman, Quantum Localization, in Quantum Dynamics of Simple Systems, Proc. of the 44-th Scottish Universities Summer School in Physics, Stirling, Aug. 1994, Edited by G.L. Oppo, S.M. Barnett, E. Riis and M. Wilkinson.

Figure 5: The dependence ξ(K) in the quantum standard map.

D.L. Shepelyansky, Phys. Rev. Lett., 56, 677 (1986).

qlD

Figure 6

A. Backer, R. Ketzmerick and A.G. Monastra, Phys. Rev. Lett. 94, 054102 (2005)