ANDERSON Localization

It is nice to know that the computer understands the problem. But I would like to
understand it, too.

EUGENE P. WIGNER
Physics Today 46 (1993) 38

The kicked rotor plays a key role in the field of quantum chaos. On the one hand the 1979 paper by CASATI, CHIRIKOV, IZRAELEV and FORD [CCIF79] marks the starting point of the systematic analysis of the quantum dynamics of classically chaotic dynamical systems; in this study characteristic differences between the classical and quantum behaviours of the same model systems were found for the first time. In particular, the numerical investigation of the quantum kicked rotor for certain parameter values clearly yielded localized quantum states, whereas the classical counterpart of the system exhibited unbounded diffusive dynamics with respect to the energy in sharp contrast to the quantum result that established an upper limit for the energy.

On the other hand, the theoretical explanation of this observation was given in 1982 by FISHMAN, GREMPEL and PRANGE [FGP82,GFP82,PGF83], who showed that the quantum kicked rotor can be mapped onto a well-known model of solid state physics. This model, the ANDERSON model [And58,And61,And78], is characterized by pronounced localization phenomena of the quantum states, provided certain conditions are met that I discuss below. Localization in the ANDERSON model thus carries over to the quantum kicked rotor and provides the desired explanation.

In the present chapter I analyze the nonresonant (with respect to $T$) quantum kicked harmonic oscillator along lines similar to those sketched above with respect to the quantum kicked rotor. In this way, by making contact between the quantum kicked harmonic oscillator and the ANDERSON model, I provide a novel theoretical explanation for the suppressed diffusion that is numerically found to be typical for the quantum dynamics in the absence of stochastic webs as discussed in the previous chapter.

Section 5.1 is dedicated to the exposition of the theory of ANDERSON localization in its well-established area of application, the quantum kicked rotor. In this context I also briefly summarize the most important properties of the ANDERSON model of disordered solids, as far as these properties are needed for the present discussion. In section 5.2 I present numerical evidence for localization phenomena in the quantum kicked harmonic oscillator for nonresonant values of $T$, i.e. in the absence of quantum stochastic webs as discussed in the previous chapter. In section 5.3 I then show analytically that the nonresonant quantum kicked oscillator exhibits ANDERSON localization.