Although classically the kicked rotor (5.1) and the kicked harmonic oscillator (1.17) are fundamentally different, for example in that the kicked rotor is characterized by one frequency only (namely the frequency of the kick) whereas the kicked oscillator has two frequencies (the second being the frequency of the unperturbed harmonic dynamics), the quantum localization phenomena in both models can be explained by the same mechanism. This mechanism is described by the theory of ANDERSON localization as outlined in section 5.1. I now show that this theory -- with some alterations due to the differing eigenstates of the two model systems -- can also be applied to the quantum kicked harmonic oscillator.
As in equation (5.43) I begin with the quantum
map (2.37) for the kicked harmonic oscillator,
Projecting equation (5.77) onto the eigenstates
of the free harmonic oscillator (cf. equation (2.38)), I get
The first remark concerns the diagonal energies.
As their rotor counterparts,
can be viewed as being generated by a
number generator that follows a Lorentzian distribution
condition of nonresonance is satisfied. Because of the different
expressions (5.58) and (5.81)
defining the respective diagonal energies, the (non-) resonances in
question differ, too. For the quantum kicked oscillator, the
inhibiting quasi-randomness of the are given by
The fact that the resonance condition (1.22/4.21a) -- and thereby the important special case (1.23/4.22) as well --, which is essential for the formation of classical and quantum stochastic webs, naturally arises as equation (5.85) in this quantum theory and characterizes those situations where no quantum localization can be expected, is a strong argument in favour of this formulation of the theory of localization in the kicked harmonic oscillator: in this way proper quantum-classical correspondence is automatically guaranteed. What is more, in this way also the quantum theories for the cases of resonance and nonresonance nicely fit together in a complementary fashion. The resonances with respect to are true quantum resonances in the sense that they are obtained here as a consequence of a genuinely quantum mechanical theory. On the other hand, as discussed above, these resonances play an important role classically as well, such that the term ``quantum resonance'' might be questioned. This situation is to be compared with the quantum resonances (5.67) of the kicked rotor and (4.7, 4.44, 4.49) of the resonant kicked harmonic oscillator which have no classical counterpart.
The second important difference to the case of the rotor is that
the oscillator the evaluation of the
hopping matrix elements
is not as simple as in
since the oscillator eigenfunctions
(2.39) are algebraically much more
complicated than the simple exponentials of the rotor eigenfunctions
(5.12). Explicitly, the matrix elements
for the cosine kick potential (1.18) are given
It is obvious that, because of the dependence on the HERMITE polynomials, the formula (5.86) cannot be simplified to yield an expression that depends on a single index only as the corresponding rotor matrix element (5.54): the matrix elements have to be viewed as functions of both the position and the difference . This means that the interaction between the sites of the ANDERSON lattice for the oscillator is not translation invariant; in other words, the strength of the interaction between two sites does not depend on their distance on the lattice alone. For the proof of ANDERSON localization this is not an obstacle, as long as for all the absolute values of the matrix elements decay rapidly enough with the distance from the site , in much the same way as the absolute values of the rotor matrix elements do.
Analytically not much can be said in general about the values
takes on. The only straightforward property to notice is that every
second of them vanishes,
The convergence for larger values of is checked in figures 5.18 and 5.19; since the integral in equation (5.86) cannot be solved analytically in general, the matrix elements are evaluated numerically and plotted for several parameter combinations. Due to the oscillatory integrand, it is difficult to obtain exact results by numerical computation for some of these parameter combinations. This difficulty has an impact especially on the results for larger values of and becomes worse in the case of small values of , as can be seen in the figures: the data for and in figures 5.18c and 5.19c are spoiled by numerical round-off errors, and the same appears to be true for the points with and or in figure 5.19a. In the opposite case of small and large the numerical results nicely agree with the convergence to zero as predicted by the relation (5.93).
Apart from the numerical problems, the figures give a clear indication of the typical behaviour of the matrix elements. The are peaked around and decay more or less exponentially on both sides. Figure 5.19 also illustrates that in general for any site the speed of this decay is not the same for and , in agreement with the inequality (5.87). By taking suitable limits (with respect to the parameters), the decay fast enough to allow equation (5.84) to approach a tight binding equation. Some aspects of the case of those parameter combinations which do not give rise to tight binding are addressed in subsection 5.3.4 below.
This leads to another observation which marks a noteworthy difference to
the rotor. Above, I have discussed which conditions have to be met for
equation (5.84) to become
a tight binding equation.
Strictly speaking, equation
(5.84) with the cosine kick potential
cannot become a tight binding equation under any circumstances, because
equation (5.88) indicates that no site is coupled
to its nearest neighbour, and the closest interacting sites are
characterized by indices , with .
Even more, the dynamics
on the sites with even indices and on the sites with odd indices
This means that the cosine-kicked oscillator
be described not by
but by a pair of discrete SCHRÖDINGER equations which are
but independent of
In each of these two subsystems tight binding is possible within the approximations discussed above. Of course, if one has tight binding in both subsystems at the same time, then the entire system is tightly bound and prone to localization.
On the other hand, by choosing a different kick potential
as a replacement for the cosine potential (1.18),
The alternate kick potential (5.95) is intended here for nothing more than demonstrating how a tight binding system describing the quantum kicked harmonic oscillator can be constructed; in particular I do not discuss the classical dynamics of the harmonic oscillator with this kick potential here. However, motivated by the results described in [Jun95], one might speculate about the existence of classical -- and perhaps quantum mechanical? -- stochastic webs even for this aperiodic kick potential (in cases of resonance with respect to ). In the following I do not discuss the alternate system specified by the potential (5.95) any further.
In this subsection I have shown that the quantum kicked harmonic oscillator can be modelled by a discrete SCHRÖDINGER equation which is similar to the model used in ANDERSON's theory. Provided certain conditions are met, it is also possible to obtain an approximative tight binding model as in the case of the rotor. In the following two subsections I discuss how these findings can be used to prove ANDERSON localization in the oscillator.