Kicked rotator
It describes a free rotating stick (with moment of inertia
) in an inhomogeneous "gravitation like" field that is periodically switched on in short pulses.
Theses equations show that between two consecutive kicks, the rotator simply moves freely: the momentum
On the other hand, during each kick the momentum abruptly jumps by a quantity
an arbitrary integer) will have the same exact stroboscopic dynamics, but with dimensionless momentum shifted at any time by
(this is why stroboscopic phase portraits of the kicked rotator are usually displayed in a single momentum cell
The kicked rotator is a prototype model used to illustrate the transition from integrability to chaos in Hamiltonian systems and in particular the Kolmogorov–Arnold–Moser theorem.
, chaotic unstable orbits are no longer constraints by invariant tori in the momentum direction and can explore the full phase space.
At long time enough, the particle as thus been submitted to a series of kicks with quasi-random amplitudes.
This quasi-random walk is responsible for a diffusion process in the momentum direction
Assuming that kicks are randoms and uncorrelated in time, the spreading of the momentum distribution writes
Corrections coming from neglected correlation terms can actually be taken into account, leading to the improved expression[3]
The dynamics of the quantum kicked rotator (with wave function
As for classical dynamics, a stroboscopic point of view can be adopted by introducing the time propagator over a kicking period
(a product of two operators, one diagonal in momentum basis, the other one diagonal in angular position basis) allows to easily numerically solve the evolution of a given wave function using split-step method.
It has been discovered[1] that the classical diffusion is suppressed in the quantum kicked rotator.
There is a general argument[9][10] that leads to the following estimate for the breaktime of the diffusive behavior Where
The quantum kicked rotor can actually formally be related to the Anderson tight-binding model a celebrated Hamiltonian that describes electrons in a disordered lattice with lattice site state
In the quantum kicked rotator it can be shown,[11] that the plane wave
The full mapping to the Anderson tight-binding model goes as follow (for a given eigenstates of the Floquet operator, with quasi-energy
Dynamical localization in the quantum kicked rotator then actually takes place in the momentum basis.
If noise is added to the system, the dynamical localization is destroyed, and diffusion is induced.
The proper analysis requires to figure out how the dynamical correlations that are responsible for the localization effect are diminished.
The same calculation recipe holds also in the quantum mechanical case, and also if noise is added.
is zero (due to long negative tails), while with the noise a practical approximation is
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
In the first works [15][16] a quantum-optic type interaction has been assumed that involves a momentum dependent coupling.
The first experimental realizations of the quantum kicked rotator have been achieved by Mark G. Raizen group[18][19] in 1995, later followed by the Auckland group,[20] 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.
