Localization Established

As discussed in the previous subsection, both discrete SCHRÖDINGER equations (5.94a, 5.94b) in a natural way give rise to the approximating tight-binding equation

$$\left( \epsilon_m + W_{m,m} \right) \overline{u}_{E,m} + W... ...ne{u}_{E,m-2} + W_{m,m+2} \, \overline{u}_{E,m+2} \; = \; 0,$$ (5.73)

provided the $W_{m,m\pm 2m''}$ tend to zero with growing $\vert m''\vert$ fast enough. I now proceed to the discussion of an appropriate reformulation of this equation by means of transfer matrices. It is straightforward to apply the results to the alternate system described by equation (5.100). It is also possible to generalize this reformulation to discrete SCHRÖDINGER equations that cannot be described by a tight binding equation (5.101) because of matrix elements $W_{m,m+2m''}$ decaying only slowly with $m''$; in these cases more terms of the series in equations (5.94) need to be considered. I come back to this important point in subsection 5.3.4 below.

In terms of the transfer matrix formalism of subsection 5.1.2, the definitions

$$\cal{T}_m^{(1)} \; := \; \left( \begin{array}{cc} \displa... ...W_{m,m-2}} {W_{m,m+2}} \\ [0.6cm] 1 & 0 \end{array} \right)$$ (5.74)

and

$$\vec{u}_{E,m}^{\, (1)} \; := \; { \overline{u}_{E,m} \choose \overline{u}_{E,m-2} }$$ (5.75)

in conjunction with the equation of motion

$$\vec{u}_{E,m+2}^{\, (1)} \; = \; \cal{T}_m^{(1)} \, \vec{u}_{E,m}^{\, (1)}$$ (5.76)

seem to suggest themselves for modelling equation (5.101).

However, this formulation is disadvantageous for several reasons. First, it rests on the assumption that $W_{m,m+2}\neq 0\,$ for all $m$, because $W_{m,m+2}$ makes up the denominators of two of the matrix elements of the transfer matrix (5.102). Unfortunately, this assumption cannot be checked analytically for all combinations of $V_0$ and $\hbar$, because a closed solution of the integral in equation (5.86) is not available. Already a single $\widetilde{m}$ with $W_{\widetilde{m},\widetilde{m}+2}=0$ could spoil the whole theory. Even more,

$$W_{m,m\pm 2} \; \neq \; 0 \quad \forall\; m$$ (5.77)

is a necessary condition^5.11for the tight binding equation (5.101) to provide a meaningful model of the kicked harmonic oscillator. For later reference, I call those systems (i.e. those combinations of the parameters $V_0$ and $\hbar$) not satisfying the condition (5.105) pathological. While in the course of the computations leading to figures 5.18 and 5.19 I have not encountered any numerical evidence for such an $m$ contradicting (5.105), this is still a matter of principle and should be addressed as such. Figure 5.20

shows some examples of the typical oscillatory behaviour of $W_{m,m pm 2}$ found by numerical integration; the numerical data plotted in the figure is accurate enough to be sure that even those matrix elements which come close to zero are nevertheless clearly nonzero.

The workaround for the case of a vanishing (second-) nearest neighbour matrix element sketched in the footnote on page only works in the context of the kicked rotor, because of the translation invariance of the discrete SCHRÖDINGER equation for that model. This is in contrast to the translation noninvariance of the discrete SCHRÖDINGER equation of the kicked harmonic oscillator, i.e. the explicit $m$-dependence of the oscillator's hopping matrix element $W_{m,m+2}$ for the interaction with the nearest neighbour at the $m$-th site. Divisions by $W_{m,m pm 2}$ occur in all transfer matrix models of the kicked harmonic oscillator, including those to be discussed below, and, as a matter of principle, cannot be avoided. In which sense the condition (5.105) can be taken to be satisfied is addressed in some more detail in subsection 5.3.4.

A second, and more important, disadvantage is that the transfer matrices of equation (5.102) do not satisfy the unimodularity requirement of FURSTENBERG's theorem:

$$\det\left(\cal{T}_m^{(1)}\right) \; = \; \frac{W_{m,m-2}}{W_{m,m+2}} \; \neq \; 1$$ (5.78)

in general, as a consequence of the asymmetry (5.87) of the matrix elements of the kicked harmonic oscillator. As a result, if these particular $\cal{T}_m^{(1)}$ were used, FURSTENBERG's theorem would not be applicable and nothing could be said about the asymptotic behaviour of the $\vert\overline{u}_{E,m}\vert$.

Finally, backpropagation towards the left cannot simply be accomplished as in the case of the rotor by using the same transfer matrices $\cal{T}_m^{(1)}$ as for propagation to the right and setting

$$\vec{v}_{E,m}^{\, (1)} \; := \; { \overline{u}_{E,m-2} \choose \overline{u}_{E,m} },$$ (5.79)

because generally this gives

$$\vec{v}_{E,m-2}^{\, (1)} \; \neq \; \cal{T}_m^{(1)} \, \vec{v}_{E,m}^{\, (1)}.$$ (5.80)

The last-mentioned disadvantage is shared by all transfer matrix models of the kicked harmonic oscillator. Essentially, this is again a consequence of the asymmetry (5.87).

In the light of these shortcomings of the model specified by the matrix (5.102), the question needs to be addressed if and how the localization results of subsection 5.1.2 carry over to the oscillator.

One way to approach this problem is to define the more general vectors

$$\vec{u}_{E,m}^{\, (2)} \; := \; { \overline{u}_{E,m} \choose \Theta_{m-2} \, \overline{u}_{E,m-2} }$$ (5.81)

with yet undetermined constants $\Theta_m\in\mathbb{C}$, and to construct new $\cal{T}_m^{(2)}$ in such a way that they are unimodular from the beginning. After a little algebra one finds that

$$\cal{T}_m^{(2)} \; := \; \left( \begin{array}{cc} \displa... ...ac{1}{\Theta_m} \\ [0.6cm] \Theta_m & 0 \end{array} \right)$$ (5.82)

are suitable unimodular transfer matrices if the $\Theta_m$ satisfy the recurrence relation

$$\Theta_m \; = \; \frac{W_{m,m+2}}{W_{m,m-2}} \; \Theta_{m-2} \, ;$$ (5.83)

the choice of the initial conditions $\Theta_0$ and $\Theta_1$ is free. Using $W_{m,m+2}=W_{m+2,m}$, and choosing $\Theta_0=W_{20}$ and $\Theta_1=W_{31}$,

$$\Theta_m \; = \; W_{m,m+2}$$ (5.84)

is obtained. No explicit closed formula for the $\Theta_m$ can be given because the same is true for $W_{m,m+2}$. In this way I obtain

$$\vec{u}_{E,m}^{\, (2)} \; = \; { \overline{u}_{E,m} \choose W_{m-2,m} \, \overline{u}_{E,m-2} }$$ (5.85)

and

$$\cal{T}_m^{(2)} \; = \; \left( \begin{array}{cc} \display... ...1}{W_{m,m+2}} \\ [0.6cm] W_{m,m+2} & 0 \end{array} \right).$$ (5.86)

Equations (5.113, 5.114) and the dynamical equation

$$\vec{u}_{E,m+2}^{\, (2)} \; = \; \cal{T}_m^{(2)} \, \vec{u}_{E,m}^{\, (2)}$$ (5.87)

combined provide an equivalent reformulation of the tight binding equation (5.101); they are a meaningful approximation to the discrete SCHRÖDINGER equation (5.94) as long as the hopping matrix elements for nearest neighbour interaction are all nonzero (``nonpathological'') and the $W_{m,m\pm 2m''}$ decay fast enough with growing $\vert m''\vert$.

For proving localization in the kicked harmonic oscillator it suffices to consider the transfer dynamics in positive $m$-direction, because the lattice on which the discrete SCHRÖDINGER equation of the oscillator is defined is infinite towards the right only, as opposed to the bi-infinite lattice in the case of the rotor. So localization towards the left is automatically built into the system by the condition $m\geq 0$, once FURSTENBERG's theorem has been shown to apply. Nevertheless it is instructive to establish the analogy with the conventional tight binding models as far-reaching as possible. In fact, using the models discussed below, exponential localization is typically found in both directions -- cf. figures 5.12a, 5.12b. What is more, a discrete point spectrum of energy eigenvalues describing the localized dynamics can only be expected in the case of exponential localization in both directions -- cf. subsection 5.1.2.

In the rotor case, the same transfer matrix can be used for right- and leftward dynamics on the lattice, provided the two different sets of vectors $\vec{u}_{E,m}$, $\vec{v}_{E,m}$ of equations (5.62) are used. For the oscillator, on the other hand, the nontriviality of the off-diagonal matrix elements of the $\cal{T}_m^{(2)}$ necessitates the introduction of another set of transfer matrices for leftward transfer. No vectors $\vec{u}_{E,m}^{\, (2)}$, $\vec{v}_{E,m}^{\, (2)}$ could be constructed that allow to model the tight binding equation (5.101) using the same matrix for propagation into both directions. In a sense, the situation in the oscillator case is reversed, as compared to the rotor: the simplest formulation of backpropagation is obtained by using the same vectors $\vec{u}_{E,m}^{\, (2)}$ as for propagation in positive $m$-direction, but another set of transfer matrices, namely (quite naturally) the inverses of the $\cal{T}_m^{(2)}$:

$$\vec{u}_{E,m-2}^{\, (2)} \; = \; \cal{T}_m^{\rm (2,left_1)} \, \vec{u}_{E,m}^{\, (2)}$$ (5.88)

with

$$\cal{T}_m^{\rm (2,left_1)} \; := \; \left( \cal{T}_{m-2}^{(... ...silon_{m-2}+W_{m-2,m-2}}{W_{m-2,m}} \;\; \end{array} \right).$$ (5.89)

Alternatively, if for the sake of analogy one wants to use the vectors

$$\vec{v}_{E,m}^{\, (2)} \; := \; { W_{m-2,m} \, \overline{u}_{E,m-2} \choose \overline{u}_{E,m} }$$ (5.90)

then one has

$$\vec{v}_{E,m-2}^{\, (2)} \; = \; \cal{T}_m^{\rm (2,left_2)} \, \vec{v}_{E,m}^{\, (2)}$$ (5.91)

with

$$\cal{T}_m^{\rm (2,left_2)} \; := \; \left( \begin{array}{cc... ... \displaystyle \frac{1}{W_{m-2,m}} & 0 \end{array} \right).$$ (5.92)

Summarizing, the transfer matrices $\cal{T}_m^{(2)}$, $\cal{T}_m^{\rm (2,left_1)}$ and $\cal{T}_m^{\rm (2,left_2)}$ can be used to describe the right- and leftward tight-binding dynamics on the lattice in all but the pathological cases. Up to this point no statement has been made concerning the respective localization properties that might follow from this description. In the following I discuss just this aspect of the dynamics.

The above construction of a tight-binding model for the kicked harmonic oscillator is made to be as close to the corresponding model for the kicked rotor as possible. However, comparing equations (5.114, 5.117, 5.120) with their rotor counterpart (5.61) makes it clear that the harmonic oscillator case is considerably more intricate, due to the presence of $W_{m,m}$ and $W_{m,m+2}$ in the matrix elements. This makes the application of FURSTENBERG's theorem more difficult than in the canonical rotor case.

In many cases, the transfer matrices $\cal{T}_m$ defined in equations (5.114, 5.117, 5.120) satisfy the assumptions of FURSTENBERG's theorem: the $\cal{T}_m$ are all unimodular by construction, the respective groups generated by the sets $\left\{ \cal{T}_m, \; m\in\mathbb{N}\right\}$ are noncompact and irreducible in the sense of appendix B, and all available numerical evidence supports the assumption that the $\cal{T}_m$ considered give rise to ``sufficiently well-behaved'' measures $\mu$ as defined by the integrability condition (B.2).

The last remark may be made more explicit by discussing in some more detail the assumptions of FURSTENBERG's theorem for the $\cal{T}_m$ considered here. From appendix B, the integrability condition is

$$\int\limits _{G} {\mbox{d}}\mu(M) \, \log(\Vert M\Vert _{\rm m}) \; < \; \infty ,$$

where $G$ is the matrix group to which the $\cal{T}_m$ belong, $\mu$ is the measure according to which the $\cal{T}_m$ are distributed in this group, and $\Vert M\Vert _{\rm m}=\max\limits_{i,j}\vert M_{ij}\vert$ is the maximum norm of the matrix $M$. For more details on FURSTENBERG's theorem see appendix B.

In order to establish the validity of the condition (B.2) with respect to the maximum norm, the distributions of values of the matrix elements of the $\cal{T}_m$ need to be checked. The off-diagonal matrix elements of $\cal{T}_m^{(2)}$, $\cal{T}_m^{\rm (2,left_1)}$ and $\cal{T}_m^{\rm (2,left_2)}$ are determined by the nearest neighbour interactions $W_{m,m+2}$, for which no closed formula is available, as discussed earlier. Nevertheless, the corresponding distribution of values can be studied numerically. Figures 5.21b, 5.21c and

5.22b, 5.22c

present distributions of $\big(\cal{T}_m^{(2)}\big)_{12}=-1/W_{m,m+2}$ and $\big(\cal{T}_m^{(2)}\big)_{21}= W_{m,m+2}$ for $V_0=0.01$ and two different values of $\hbar$. Obviously, the distributions are nicely peaked, and make it thus plausible that, by integration in the sense of (B.2), they contribute to $\cal{T}_m^{(2)}$ satisfying the integrability condition. It would be desirable to strengthen this argument by considering other parameter combinations as examples as well, but unfortunately the numerical effort for this task is very high; already the computation of the matrix elements needed for figures 5.21 and 5.22 required a considerable amount of computer time. Still, it is natural to assume that a similarly unproblematic behaviour of the $W_{m,m+2}$ is given for other values of $V_0$ and $\hbar$ as well. A more stringent reasoning leading to (B.2) being satisfied is presented in subsection 5.3.3 below.

It remains to investigate the nontrivial diagonal matrix elements of the $\cal{T}_m$, namely

$$\Big( \cal{T}_m^{(2)} \Big)_{11} \; = \; \left( \cal{T}_{m... ...} \right)_{11} \; = \; -\frac{\epsilon_m+W_{m,m}}{W_{m,m+2}}.$$ (5.93)

Since these are functions not only of the hopping matrix elements $W_{m,m'}$, but also of the diagonal energies $\epsilon_m$, the results depend on the parameters $E$ and $T$, too, and general analytical conclusions become even more difficult. On the other hand, $E$ and $T$ enter into the matrix elements only via the (quasi-) random number generator (5.81), and as long as $T$ takes on nonresonant values, the actual values of the two parameters should be expected to be more or less irrelevant. One can at least expect that the familiar Lorentzian shape of the distribution of $\epsilon_m$ (cf. figure 5.17a) does not get distorted too much by the combination with $W_{m,m}$ and $W_{m,m+2}$ in equation (5.121). For the particular case shown in figure 5.17a (with $E=1.0$, $T=1.0$, $\hbar =1.0$), this expectation is verified in figure 5.21a, where a histogram for the distribution of the $\big( \cal{T}_m^{(2)} \big)_{11}$ is shown; figure 5.22a shows the same for $\hbar =0.1$.^5.12The figures make it clear that -- at least for these combinations of parameters, but probably in a more general sense -- the values of the $\big( \cal{T}_m^{(2)} \big)_{11}$ are still nicely peaked around a finite value and likely give rise to a smooth and sufficiently fast decaying $\mu$. Within the accuracy of the histograms, the distributions of the matrix elements involving the diagonal energies even take on quite exactly the form of a Lorentzian, for which compliance with (B.2) is analytically shown in appendix B. But note that the distributions are broad, with half widths of the order of magnitude of $10^3$ and $10^2$, respectively, obviously a consequence of the small values the $W_{m,m+2}$ take on.

Summarizing, although a rigorous proof is lacking, there is good reason -- including some convincing numerical evidence -- to assume that typically the condition (B.2) is satisfied for the matrices $\cal{T}_m^{(2)}$, $\cal{T}_m^{\rm (2,left_1)}$ and $\cal{T}_m^{\rm (2,left_2)}$ modelling the kicked harmonic oscillator.

As a result, I have shown that at least for some combinations of values of the parameters, FURSTENBERG's theorem can be applied to the transfer matrix formulation of the kicked harmonic oscillator in a way which is largely analogous to the conventional procedure in the case of the kicked rotor; it may be assumed that similar results probably hold for many other values of the parameters. By the same reasoning as in subsection 5.1.3 it can be concluded that the norms of the vectors $\vec{u}_{E,m}^{\, (2)}$ and $\vec{v}_{E,m}^{\, (2)}$ generated by the respective equations of motion decay exponentially, which in turn implies the same for the absolute values of the expansion coefficients $\vert\overline{u}_{E,m}\vert$ of the quasienergy states discussed in subsection 5.3.1. This finally establishes the result that the nonresonant kicked harmonic oscillator exhibits ANDERSON localization, provided the parameters are such that the conditions are met which I have discussed above. When this is the case then generic sequences of states $\left\vert \psi_n \right>$ generated by the quantum map (5.72) consist of exponentially localized states. The states are localized with respect to the basis of harmonic oscillator eigenstates, and therefore they are localized in phase space as well. Figure 5.12 stresses the first aspect of localization; figures 5.9 through 5.11 and the examples in section C.4 of the appendix demonstrate localization in phase space.

As discussed in subsection 5.1.2, a lower bound for the speed of the decay of $\vert\overline{u}_{E,m}\vert$ is established via the LIAPUNOV exponent of the corresponding transfer matrices. From this point of view it does not come as a surprise that in the case of the oscillator the transfer matrices $\cal{T}_m^{(2)}$ and $\cal{T}_m^{\rm (2,left_1)}$ or $\cal{T}_m^{\rm (2,left_2)}$ do not coincide as they do in the case of the rotor. This is again a consequence of the asymmetry expressed by the inequality (5.87) and can lead to different speeds of convergence to zero of $\vert\overline{u}_{E,m}\vert$ on the right and on the left. But note that FURSTENBERG's theorem provides a lower bound on the speed of convergence only; the speed may be actually larger, and in special cases it can coincide on the right and on the left.

Inevitably, the above discussion of the distribution of values of the matrix elements -- especially of the off-diagonal matrix elements -- of the transfer matrices is to some degree heuristic, and therefore the same is true, unfortunately, with respect to the distribution $\mu$ of the transfer matrices. In particular the arguments in favour of nicely distributed off-diagonal matrix elements leading to FURSTENBERG-integrability are but qualitative. This problem is addressed -- and solved -- in subsection 5.3.3 by considering a different class of transfer matrices.

Footnotes

... condition^5.11

Note that $W_{m,m+2} \neq 0 \; \forall m$ implies $W_{m,m\pm 2} \neq 0 \; \forall m$, because $W_{m,m'}=W_{m',m}$.

....^5.12

Note that due to the intricacy of the numerical evaluation of the $W_{m,m'}$, making it very computer time consuming, only a much smaller number of values of the $\big( \cal{T}_m^{(2)} \big)_{11}$ could be taken into account here than in figure 5.17a ($10^3$ values versus $10^5$ values), thus giving rise to the quite coarse-grained histograms in figure 5.21 and 5.22.