Mapping of the Quantum Kicked Harmonic

Oscillator onto an ANDERSON-like Model

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,

which are similarly defined as their rotor counterparts in equations (5.45) and (5.47). Using the quasienergy states before and after the kick, the quantum map (5.72) becomes

With the operator

I transform the kick equation (5.74a) into

which in terms of the averaged quasienergy state gives

by combination with the equation (5.74b) describing the free part of the dynamics. As in the case of the kicked rotor the original dynamical system given by equation (5.72) is thus mapped onto a static problem by exploiting the FLOQUET character of the system.

Projecting equation (5.77) onto the eigenstates
of the free harmonic oscillator (cf. equation (2.38)), I get

(5.52) |

Collecting terms I then arrive at the equation

with the diagonal energies

Completeness of implies that can be expanded according to

(5.56) |

have been introduced. Summarizing, the discrete SCHRÖDINGER equation that describes the quantum dynamics of the kicked harmonic oscillator has been derived:

The argument list of the is dropped for convenience, as usual. But note that here -- in contrast to the diagonal energies of the rotor (5.58) -- the depend on the kick period as well. In the context of localization, this is a very important observation, as will be seen shortly when the issue of resonances is discussed. While the above derivation at first sight seems to be entirely analogous to the situation in subsection 5.1.3, there are several important differences that I discuss in the following.

The first remark concerns the diagonal energies.
As their rotor counterparts,
the
can be viewed as being generated by a
(quasi-)
random
number generator that follows a Lorentzian distribution
(5.70),
provided a
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
*quantum resonances*
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
for
the oscillator the evaluation of the
hopping matrix elements
is not as simple as in
subsection 5.1.3,
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
(5.83)
for the cosine kick potential (1.18) are given
by^{5.10}

while for the rotor holds.

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,

(5.63) |

According to [GR00] the integral can be evaluated in the following way: (For notational convenience, the following formulae up to (5.93) are meant for positive values of only; similar formulae hold for negative and large enough .)

(5.64) |

and the generalized LAGUERRE polynomials satisfy the inequality [AS72]

(5.65) |

(5.66) |

which shows that the absolute values of the matrix elements decay more than exponentially fast with the distance ; in addition, as in the case of the rotor the amplitudes of all are controlled by the factor and can thus be made as small as desired by taking the limit . Summarizing, the inequality (5.93) shows that for weak perturbations and for not too large values of the discrete SCHRÖDINGER equation (5.84) with the cosine kick potential (1.18) indeed approaches a tight binding equation.

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
are
completely decoupled.
This means that the cosine-kicked oscillator
must
be described not by
a single,
but by *a pair of* discrete SCHRÖDINGER equations which are
interwoven with
but independent of
each other:

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),

corresponding to (cf. equation (5.75)) are given by

(5.69) |

(5.70) |

(5.71) |

coupling each site to its nearest neighbour sites. Note that for the kicked harmonic oscillator no alternate potential could be constructed that yields a tight binding system coupling each site to its respective second-nearest neighbours only, as motivated by equations (5.94).

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.

- ...
by
^{5.10} - For the integral in (5.86) is not RIEMANN integrable, because in this case the argument of the tangent can take on values equal to half-integer multiples of , making the integrand singular. However, with probability one (with respect to the values of and ), the still take on well-defined values if the integral is evaluated as the corresponding CAUCHY principal value of (5.86). Therefore this issue does not challenge the proper physical interpretation of the discrete SCHRÖDINGER equation (5.84).