Kicked rotator and dynamical localization

The model of kicked rotator was introduced by Casati, Chirikov, Ford and Izrailev in 1979 and became one of the basic model of quantum chaos. In fact this model is obtained by quantization of the Chirikov standard map. The numerical simulations 1979 showed that the classical chaotic diffusion is completely suppressed by quantum effects.

For a better understanding of quantum dynamics of classically chaotic systems the concept of two principal time scales was introduced in [4]. The first (Ehrenfest) time scale $t_{E}$ is proportional to the logarithm of a typical quantum number $q \sim 1/\hbar$ ( $t_{E} \sim \ln q$). For times shorter than this scale the minimal coherent wave packet completely follows a chaotic classical trajectory and according to the Ehrenfest theorem there is complete agreement between classical and quantum mechanics. However, due to local exponential instability of classical motion the packet is fastly destroyed. However, the classical diffusion goes on on a much larger diffusive time scale $t_D$. The estimate for this time scale (break time) was obtained in [4] ($t_D \sim q^2$). After this time the classical diffusion is localized by quantum interference.

The Ehrenfest time scale is very short and after this scale there is no instability in quantum dynamics. Due to that the exponential decay of correlations which existed in the classical system with hard chaos (no islands of stability) is replaced by a very slow power decay or some residual constant level of correlations in the quantum system (refs. [3, 14]). This leads to the practical reversibility of quantum motion in time. Indeed, after time reversal the quantum system returns back to the initial state with computer accuracy, while the classical system due to exponential instability and computer round off errors continues to diffuse [14,21].

The analogy established by Fishman, Grempel and Prange tells that the localization of chaotic classical diffusion is analogous to the Anderson localization in one-dimensional random potential. However, in the kicked rotator the localization is of dynamical origin since the model is completely deterministic and it has no random parameters. The more detailed further investigations of this dynamical localization were carried out in [14,20,24,27]. In [20] the transfer matrix technique was for the first time applied to the problem of dynamical localization (later it was also used by Blumel and Smilansky). On the basis of this technique and analytical theory for simple models it was shown that the localization length for quasienergy eigenstates is given by the classical diffusion rate $l = D/2 \sim t_{D}$ [20,27]. Recenly, this result was confirmed by Altland and Zirnbauer, and Frahm on the basis of supersymmetry approach to the kicked rotator with random phases.

The estimate for the diffusive time scale $t_{D}$ was also obtained on the basis of Maslov formula for quasiclassical expansion of wave function over classical trajectories. The main term in the sum contains exponentially many terms due to local instability of classical motion. In his formula Maslov also gave a general expression for corrections to this term which are proportional to higher powers of $\hbar$. The analysis of these corrections [2,4] for chaotic systems shows that these corrections grow only as a power of time and gives the estimate $t_{D} \sim q^2$ in the case of kicked rotator.

The kicked rotator model was recently realized in experiments with cold atoms by M.Raizen et al.. (see also section 3.5.3).