The Kicked Rotor

The kicked rotor is one of the most frequently studied model systems in
dynamical systems theory. It emerges in many physical systems
(see for example [Zas85,MRB^{+}95]),
and despite being very simple it has successfully been used
for modelling the onset of chaos, retaining many of the typical and
complex features of the underlying
physical
system.

After suitable scaling,
leading to dimensionless variables,
the kicked rotor can be defined by the
Hamiltonian

such that models, for example, a mathematical pendulum that is driven by ``impulsively acting gravity'' [CCIF79,LL92]. The first summand of describes free rotation; the second specifies the periodic ``gravitational'' kicks the strength of which depends on the kick function. See figure 5.1 for a schematic illustration. In subsection 5.1.3 other kick functions are considered as well.

Note that while the Hamiltonian (5.1) with the harmonic forcing (5.2) is similar to the Hamiltonian (1.17) of the kicked harmonic oscillator there are two essential differences: the phase space of the rotor is a cylinder -- as opposed to the phase plane of the oscillator -- and there is no harmonic potential term like in . Notice further that the rotor depends on the single parameter only that controls the amplitude of the kicks. This is in contrast to the oscillator, where a second parameter (for example the period of the kicks) cannot be eliminated by scaling. As I outline below these seemingly minor differences account for remarkably different dynamics of the two model systems, both in classical and quantum mechanics.

Defining , as the values of ,
immediately before the -th kick,

one obtains from the discrete time dynamics

The bifurcation scenario of this map for increasing values of is a typical KAM scenario in the sense that, as is increased, more and more invariant tori, guaranteed to exist by the KAM theorem [GH83], are destroyed. Some phase portraits -- periodic with period both in the - and -directions -- of this transition to chaos are shown in figure 5.2.

For (figure 5.2a) the phase portrait is dominated by the invariant lines of regular dynamics. For the intermediate parameter value (figure 5.2b) the regime of
Energy diffusion of the rotor
in the classical -phase space can be described by
considering the rotational energy^{5.2}before the -th kick:

(5.4) |

such that normally diffusive dynamics is to be expected. This averaging is justified for large enough , when unhindered diffusion through phase space is possible. Corrections to formula (5.7b) resulting from accelerator modes and angular correlations are discussed in [LL92], for example.

In analogy with the quantum map of the kicked harmonic oscillator
introduced in section 2.1,
the quantum dynamics of the kicked rotor is
given by

(5.5) |

for the kick and the unperturbed dynamics, respectively, where and are the angle and angular momentum operators, and is the potential energy operator. Unlike the classical system that contains the single parameter only, the quantum system depends on

The quantum map (5.8) can be iterated
comparatively easily
since the unperturbed dynamics of the rotor is
*free rotation* and the propagator
becomes a mere multiplication operator in the
angular momentum representation: for
expanded according to

(5.5) |

with respect to the eigenvalues , one obtains for the free rotation part of the dynamics:

Switching between the angle and angular momentum representations can be accomplished with little numerical effort by

Results of numerical experiments for both the classical and quantum rotors are shown in figure 5.4, where classical normal diffusion in the case is contrasted with quantum mechanically suppressed diffusion.

The classical diffusion coefficient is found numerically as , a somewhat smaller value than given by the large approximation (5.7b), due to the small value of (cf. the discussion, at the end of subsection 1.2.4, of a similar situation with respect to the kicked harmonic oscillator). On the other hand, the quantum energy expectation value as a function of discretized time ,(5.6) |

It is this
quantum mechanically suppressed energy diffusion, or
*quantum localization*, of the
kicked rotor that is
to be explained in the following two subsections.

- ... map.
^{5.1} - The sign of the force term in the standard map (5.4) or in the potential (5.2) is a matter of convention. Changing this sign is equivalent to shifting by .
- ... energy
^{5.2} - Due to the boundedness of the rotor's phase space in the direction of , energy diffusion occurs only in the (angular) momentum coordinate here, as opposed to the case of the kicked harmonic oscillator, where both the position and momentum variables , are unbounded and subject to diffusion -- cf. equations (1.74, 1.77).