Energy Growth within Quantum Stochastic Webs

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
considered.
Using the BCH formula

This equation is now used to identify those translations defined by equations (4.32-4.34), respectively, that are independent of each other in the sense that they form

For these groups of commuting translation operators the obvious multiplication rule is used:

- :
For this value of it follows from equations (4.34) and (4.37) that is equivalent to

There are two ways for this equality to hold for all :

- -
- First
there is the case of

The translation operators of this type can be organized as a family -- indexed by -- of commutative*one*-parameter groups of translations. For , these groups are given by

All translations in such a group shift along the same direction in the -plane, defined by the*rational*gradient

For , the group is

and consists of vertical translations. - -
- Alternatively, equation (4.39) is satisfied
for all
if there is an
such that

i.e. if is an integer multiple of . Then the set

of all translations satisfying equations (4.34) is in fact a commutative*two*-parameter group.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 obtained from equations (4.33) and (4.37) if and only if

which can be satisfied in two ways:

- -
- Either
satisfy equation (4.40),
making this case very similar to the corresponding case for .
Again, the family of Abelian one-parameter groups obtained in this way
is indexed by the parameter
.
For
, one has the groups

and for , the group of vertical translations is

- -
- Equation (4.46) is also satisfied
for all
if there is an
such that

This being granted, with equations (4.33) the commutative two-parameter group

is obtained, which is similar to the group (4.45) for .

- :
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
,

(4.23) |

with being the average value of in . For sufficiently large , and on the assumption that covers more or less uniformly,

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,
or
;
the indices
,
label the eigenstates of
and
, respectively.
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

(4.24) |

For sufficiently large , the exponential in equation (4.59) becomes a rapidly oscillating function of , such that the integral can be evaluated using a stationary phase argument [JJ72]. Assuming without loss of generality that, for each , has a single critical point at , I have:

where for . Initial states with can be excluded on generic grounds, but even in such cases comparable results may be found [Olv74]. Similarly, the method can be generalized to cover situations with as well. The important point about equation (4.60) is that it indicates that the HUSIMI distribution at a given phase space point asymptotically decays like

with time . This in turn, together with equation (4.55b), gives

that has already been found numerically -- cf. equation (4.6). In this way, asymptotically unbounded energy growth has been proven, and this energy growth follows a linear,

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
(4.45, 4.50),
there is a complete set of common eigenstates
of
and, for example,
and
(or
and
);
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

(4.29) |

(4.30) |

As in equation (4.60), this integral is evaluated using a stationary phase argument. For each , it is assumed without loss of generality that has a single critical point at and that the Hessian at this point is either negative or positive definite, such that for negative/positive definite can be defined. Then I obtain for large enough :

In this way, as a result of the

as opposed to the previous result (4.61). This, with equation (4.55b), implies

thereby explaining the asymptotically quadratic,

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.

- ... uniformly,
^{4.4} -
As the contour plots in subsection 4.1.1
and in appendix C
show,
this assumption is somewhat ill-justified, but at least
within the family
of phase space distributions
discussed in appendix A,
is the smoothest
(cf. equation (A.92))
and thus comes closest to
satisfying
this assumption.
What is more, for the estimate (4.55a) it is
also used that the HUSIMI distribution
-- by equations (A.86) --
is non-negative.
Table A.1 shows that this advantageous feature is a
unique property of
(and thus of )
within the family
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.

- ...
as
^{4.5} - Due to the approximative nature of the derivations in this subsection it is not necessary here to consider, more exactly, the stroboscopic times just before the -th kick, rather than .