Discussion

In this subsection I discuss some of the problems of the theory developed in the previous subsections, and I outline how these problems can be either avoided or solved.

One of the issues that need to be addressed is to what degree quantum localization can be expected when the matrix elements $W_{m,m+m''}$ do not decay rapidly enough with $m''$. This problem is not unique to the quantum kicked harmonic oscillator; the same can happen with respect to the quantum kicked rotor: figure 5.7 indicates that for larger values of $V_0$ the quality of the tight binding approximation deteriorates there, too. In the literature, this is obviously not regarded as an obstacle to ANDERSON localization in the kicked rotor, based on the tight binding equation; at least, this problem is not addressed in the standard literature. But see [Kle97] for a discussion of this aspect, with a positive result on localization in a particular ANDERSON model with ``long range hopping''.

Judging from figures 5.18 and 5.19, such a case of slow decay should be expected in the quantum kicked harmonic oscillator for larger values of $V_0$ and $\hbar$. Even in situations like this, localization is generically given. This can be seen by replacing the tight binding equation (5.101) with a similar approximation to the discrete SCHRÖDINGER equation (5.84), the only difference being that now the interaction with neighbouring sites is assumed to be characterized by a suitably longer range $m''_{\rm max}$ (with even $m''_{\rm max}\geq2$):

$$\epsilon_m \overline{u}_{E,m} + \sum_{{\scriptstyle m''=-\m... ...\rm max}} \, W_{m,m+m''} \, \overline{u}_{E,m+m''} \; = \; 0.$$ (5.91)

Replacing the tight binding equation, this ``not-so-tight-binding equation'' can be used in the scheme for constructing transfer matrix models as described in subsection 5.3.3.

The same technique helps to avoid the troubles that come with the pathological cases where $W_{m,m+2}=0$ or $W_{m,m-2}=0$ for some $m$. The problem of unavoidably having to divide by such a zero, as encountered in subsections 5.3.2 and 5.3.3, simply vanishes for suitably large $m''_{\rm max}$ if there exists nontrivial interaction between some sites at all.

The other, probably more satisfying, answer to the question of the pathologically vanishing nearest neighbour interactions is the following. The matrix elements for nearest neighbour interaction can be regarded as the values of a continuous function $f(z;V_0,\hbar)$, sampled at equidistant intervals on the $z$-axis. Figure 5.20 gives the impression that $f$ is well-behaved in the sense that all its intersections with the $z$-axis are transversal, and that the zeroes of $f$ have no accumulation point on the $z$-axis. Then it follows that generically pathological cases do not occur [Str01a]. In all those degenerate cases with $W_{m,m+2}=0$ or $W_{m,m-2}=0$ for some $m$, an arbitrarily small variation of either $V_0$ or $\hbar$ should suffice to eliminate the pathological situation. The disadvantage of this argument, though, is its heuristic nature, due to the lack of an explicit, integrated formula for the $W_{m,m'}$.

ANDERSON localization of the kicked harmonic oscillator appears to be a quite general, and perhaps even generic phenomenon: in the preceding subsections, I have motivated the fact that the most important ingredients of all transfer matrices are the diagonal energies (5.81) which are independent of the kick potential. Only the period $T$ of the kicks enters equation (5.81) in a truly significant way. The matrix elements (5.83), on the other hand, have been seen to be rather irrelevant for the localization properties -- as long as they give rise to tight or at least narrow binding -- and the matrix elements mark the only point where properties of the kick potential enter the theory. What is more, in the matrix elements the kick potential effectively gets randomized by being used as the argument of the tangent only -- cf. equation (5.75). In this sense, quantum localization of the nonresonant kicked harmonic oscillator should be expected for generic kick functions, not only for the cosine kick potential (1.18). With respect to the value of $T$, one may also speak of genericity of localization in an additional sense: under variation of $T$, the resonant values (5.85) not leading to ANDERSON localization are obtained with probability zero.

It is interesting to note the fundamental difference between the conditions of nonresonance that lead to quantum localization in the two model systems considered in this chapter: The resonance condition (5.67) for the quantum kicked rotor is obviously a consequence of the quantum nature of the system, as the expression (5.67) involves the PLANCK constant $\hbar$ and has no classical counterpart. On the other hand, nonresonance in the context even of the quantum version of the kicked harmonic oscillator involves just the classical resonance condition (1.22) on the kick period $T$. It is one of the advantages of the particular scaling (1.15, 1.16, 2.4) of the dynamical variables and parameters employed here, that it clearly exposes this property of the dynamics, in contrast to the scaling used elsewhere [BRZ91]. So, using the abovementioned scaling, quantum localization in the kicked harmonic oscillator comes about as a genuine quantum phenomenon -- without a classical analogue -- that is entirely controlled by a classical parameter: the question of nonresonance of the kick period $T$ alone decides on the localization of quantum states in the kicked harmonic oscillator. In particular, the existence of localization is independent of the strength of the kicks. Then, the other parameters $V_0$ and $\hbar$, via the matrix elements (5.86), control the strength of the interaction between the ``sites'' and determine if there is -- approximately -- tight binding or rather some medium range interaction; in the same way they control the localization length of the system.

A final remark on the numerical results in section 5.2 and section C.4 of the appendix: for some of the presented examples, the quantum map has been iterated for a very large number of times. Up to and beyond $10^5$ kicks have been considered, while the number of harmonic oscillator eigenfunctions taken into account for the basis was only between 3000 and 6000. This was possible just because of the quantum localization of the systems considered in that and the present section; otherwise, for example for the resonant systems discussed in chapter 4, such basis sizes would have allowed kick numbers only typically not exceeding a few thousand.