The analysis described in the previous section can be extended in order to give a qualitative explanation of the energy growth that has been observed in subsection 4.1.2 as a result of the unbounded quantum dynamics in the channels of the quantum stochastic web. In this way, contact is made with the classical counterpart not only with respect to the symmetries of the phase portrait -- thereby taking a static point of view -- but also with respect to the most important dynamical property of the system.
The existence of a complete set of extended states implies that, with respect to almost all initial states , unbounded growth of the energy expectation value as a function of time is to be expected.
In order to check the explicit time dependence of this energy growth
the commutation properties of different translation operators must be
Using the BCH formula
For this value of it follows from equations (4.34) and
If is given by equation (4.44), then the commuting translation operators in any case form the two-parameter group (4.45), regardless of equation (4.42) being satisfied in addition or not.
For these two values of , commuting translation operators are
from equations (4.33) and
if and only if
While the above shows that the cases of 4 and 6 are essentially equivalent with respect to commutation of translation operators, the situation is substantially different for or , because here and can take on any real value. I do not discuss these cases any further.
Equations (4.44) and (4.49) represent a new kind of quantum resonance with respect to . It has no classical counterpart and is thus entirely different from the resonance condition (1.23/4.22) that concerns the parameter and plays quite the same role both classically and quantum mechanically, as discussed earlier. The consequences of the quantum resonances (4.44, 4.49) have been studied numerically in subsection 4.1.2.
In this way, for the existence of the commutative groups (4.41, 4.45, 4.47, 4.50) of translation operators, each commuting with the -th power of the FLOQUET operator, has been established. This is the setting of BLOCH's theorem [Mad78]. The associated energy growth can now be estimated qualitatively as follows.
The first step is to express the energy expectation value in terms of
the HUSIMI distribution function
corresponding to the state
With equation (2.31) and the
overcompleteness relation (A.52) of the
coherent states I obtain for all :
Integrals of this kind can be approximately evaluated by considering the phase space region significantly occupied by the HUSIMI distribution at time . From the normalization property (A.53) of ,
where in the intermediate step it has also been assumed that on the average grows isotropically in the phase plane -- a behaviour that is typical for the dynamics with : cf. figures 4.2, 4.4 and 4.6 (this behaviour is in contrast to the nonisotropic growth of for , examples of which are shown in figures C.38-C.40 in appendix C).
From here on the discussion has to distinguish between the cases of
the one-parameter groups (4.41, 4.47)
and the two-parameter groups (4.45, 4.50)
of commuting translation operators.
In the first case, there exists a complete set of common eigenstates
of and, for example,
label the eigenstates of
These states share the most important properties of the
quasienergy states (2.20).
In particular, they can be written as
analogous to the of equation (2.23). The phase in the exponential in the full states (4.56a) is essentially determined by which parallels the quasienergy in the states (2.20). Due to the completeness of , any initial state can be expanded as
For the two-parameter groups of commuting translation operators in the cases of the -resonances (4.44) and (4.49), the above reasoning can be repeated in a similar, but not identical, fashion. The differences in some details account for a result that is remarkably different from equation (4.62).
With the two-parameter groups
there is a complete set of common eigenstates
and, for example,
corresponding to the two group parameters there are now two indices
in addition to .
Parallel to equations (4.56) I now have
The initial state
The above explanation for the energy growth relies on the possibility to expand the states as in equations (4.57) and (4.64), and on the applicability of the stationary phase approximation. In particular the expansions (4.57, 4.64) are somewhat questionable; their -- at least approximate -- validity depends on details of the spectral properties of the respective operators. Nevertheless, the arguments based on these assumptions yield suggestive results, in accordance with the numerical findings in section 4.1.2. These points are discussed in some more detail in [GB93,BR95].
This chapter provides a description of the typical quantum dynamics of the kicked harmonic oscillator in the cases of resonance with . It is natural to ask in which way the quantum dynamics in the complementary cases of nonresonance -- where there are no phase space structures characterized by combined translational and rotational symmetries -- differs from the scenario of the present chapter. It might be conjectured that in the absence of resonance, without the condition (1.23/4.22) enforcing the existence of infinitely extended quantum states, the typical quantum dynamics is characterized by some kind of localization phenomenon. That conjecture can be considered to be motivated by the multitude of localization results that have been obtained for the quantum kicked rotor, for which the absence of a classical resonance condition like (1.23) is an essential feature. This question is addressed in the following chapter 5, where the localization approach to the dynamics is taken.
These observations, together with the fact that the conclusions obtained on the basis of the assumptions leading to the approximation (4.55a) obviously agree with the numerical results of section 4.1.2 (this is discussed on pages ff below), further explain the importance of the HUSIMI distribution for the analysis of the quantum dynamics of classically chaotic systems.