Kicked rotator
The kicked rotator, also spelled as kicked rotor, is a prototype model for chaos and quantum chaos studies. It describes a particle that is constrained to move on a ring (equivalently: a rotating stick). 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 Where formula_2 is the Dirac delta function, formula_3 is the angular position (for example, on a ring), taken modulo formula_4, formula_5 is the momentum, and formula_6 is the kicking strength. Its dynamics are described by the standard map  (which can be derived by integrating the Equations of motion) With the caveat that formula_5 is not periodic, as it is in the standard map. Here formula_9 and formula_10 are the momentum and position right before the formula_11-th pulse. Main properties (classical). In the classical analysis, if the kicks are strong enough, formula_12, the system is chaotic and has a positive Maximal Lyapunov exponent (MLE). The averaged diffusion of the momentum-squared is a useful parameter in characterizing the delocalization of nearby trajectories. The inductive result of the standard map yields the following equation for momentum The diffusion can  then be calculated by squaring the difference in momentum after the formula_14 kick and the initial momentum, and then averaging, yielding In the chaotic domain, the momenta at different time points may be anywhere from entirely uncorrelated to highly correlated. If they are assumed uncorrelated due to the quasi-random behaviour, the sum involving the cross-terms formula_16 is neglected. In this limit, since the first term is a sum of formula_17 terms all equalling formula_18, the momentum diffusion becomes formula_19. However, if the momenta at different time points are assumed highly correlated, the sum involving the cross-terms is  not neglected, and so it contributes more terms equalling formula_18. Altogether, there are formula_21 terms to sum, all of the form formula_22. This gives an upper bound on the momentum diffusion of formula_23. Therefore, in the chaotic domain, the momentum diffusion is between That is, the momentum diffusion in the chaotic domain has somewhere between a linear and a quadratic dependence on the number of kicks. An exact expression for formula_25 can be obtained in principle by calculating the sums explicitly for an ensemble of trajectories. Main properties (quantum). In the quantum analysis, the Hamiltonian must first be rewritten in  operator form, using the substitution formula_26 to give (in dimensionless form; A similar derivation with physical parameters are available in Casati et al.)…. The wavefunction can then be solved for using Schrödinger's equation where formula_29 is here scaled according to the period between kicks, formula_30, and the wave-vector of the driving potential, formula_31, as The wavefunction at the formula_33 kick can be expanded in terms of the momentum eigenstates, formula_34, as It can be shown that the coefficients are given recursively by Where formula_37 is a Bessel function of order formula_38. Given some set of initial conditions, it  is relatively straightforward to numerically solve the recursive equation above for all time, and substitute the calculated coefficients back into the momentum eigenstate decomposition to find the total wavefunction. Squaring this gives the time evolution of the probability distribution, thus providing a complete quantum mechanical description. Another way to calculate the time evolution is to iteratively apply the unitary Floquet operator 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. There is a general argument that leads to  the following estimate for the breaktime of the diffusive behavior Where formula_41 is the classical diffusion coefficient. The associated localization scale in momentum is therefore formula_42. 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. Recall that the diffusion coefficient is formula_43, because the change formula_44 in the momentum is the sum of quasi-random kicks formula_45. An exact expression  for formula_41 is obtained by calculating the "area" of the correlation function formula_47, namely the sum formula_48. Note that formula_49. The same calculation recipe holds also in the quantum mechanical case, and also if noise is added. In the quantum case, without the noise, the "area" under formula_50 is zero (due to long negative tails), while with the noise a practical approximation is formula_51 where the coherence time formula_52 is inversely proportional to the intensity of the noise. Consequently, the noise induced diffusion coefficient is 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 formula_54 coordinate, and is still spatially homogeneous. In the first works Experiments. Experimental realizations of the quantum kicked rotator have been achieved by the Raizen group, and by the Auckland group, and have encouraged a renewed interest in the theoretical analysis. In this kind of experiment, a sample of cold atoms provided by a Magneto-optical trap interacts with a pulsed standing wave of light. The light being detuned with respect to the atomic transitions, atoms undergo a  space-periodic conservative force. Hence, the angular dependence is replaced by a dependence on position in the experimental approach. Sub-milliKelvin cooling is necessary to obtain quantum effects: because of the Heisenberg uncertainty principle, the de Broglie wavelength, i.e. the atomic wavelength, can become comparable to the light wavelength. For further information, see. Thanks to this technique, several phenomena have been investigated, including the noticeable: