Mapping of the Quantum Kicked Rotor onto the

ANDERSON Model

Combining the results of the two preceding subsections, I now present the argument showing that the quantum kicked rotor exhibits ANDERSON localization, by deriving a mapping of the rotor model onto the ANDERSON model [FGP82,GFP82,PGF83].

I begin this discussion of the quantum map (5.8)
with
a general kick potential ,

Since the are time-independent it suffices to consider in the following the time evolution of a single quasienergy state rather than the general of equation (5.44).

For the investigation of the kicked rotor the states
are much better suited than the full quasienergy states
,
since by the periodicity (2.24) the time
argument can be dropped altogether. This allows to map
the dynamical system
(5.43) onto a static problem.
In accordance with the discrete nature of the kick, I define the
(reduced) quasienergy states immediately before and after the kicks as

It turns out that the kick part (5.46a) of the dynamics is
most conveniently described using the averaged state

This definition of can be used to rewrite the kick propagator as

(5.35) |

Inserting
these expressions into the equation (5.46b)
that describes
the free part of the dynamics, the
are
eliminated from the equation and the dynamics is entirely formulated in
terms of the averaged quasienergy state
:

(5.35) |

and obtain

(5.37) |

It remains to evaluate the matrix element
.
By virtue of the representation (5.12) of
it is easily seen that the matrix element
satisfies

(5.39) |

(5.40) |

In the end one thus arrives at an equation analogous to the discrete
SCHRÖDINGER equation (5.20) with ,

the hopping matrix elements describe the coupling of a ``site of the lattice'' with its -th nearest neighbour, due to the kick. By the property (5.54) this coupling depends on the distance between the two sites only, which means that the system is translation invariant.

Note that in general
equation (5.57)
is *not*
a tight binding equation
(5.31), since the interaction is not restricted to just the
respective nearest neighbours of each site.
In fact, the cosine potential (5.2) leads to

obtained in this limit can be described -- in analogy to the transfer matrix (5.33) in ANDERSON's theory -- by the transfer matrix

acting on the vectors

via

-- cf. the vectors (5.32, 5.36) and the ``equations of motion'' (5.34, 5.37) of the ANDERSON model in subsection 5.1.2.

By choosing a different kick potential the transition to a
true tight binding model may be achieved. The choice of the potential
affects the hopping matrix elements only and leaves the diagonal energies
unchanged. For the alternate potential

(5.45) |

For
all
practical purposes the diagonal energies of
(5.58)
can be taken to be randomly distributed as in the
ANDERSON-LLOYD model, which can be seen as follows.
In the absence of the
*quantum resonances* [Fis93]
defined by

of the tangent in equation (5.58) is pseudo- or quasi-random

(5.49) |

Figure 5.8a confirms this generic distribution of the in the case of and . Figure 5.8b, on the other hand, shows the distribution of the for the just slightly detuned value ; in this particular example of quantum resonance, due to the implicit modulo operation in equation (5.58) takes on only 18 different values, as opposed to the continuous range of values obtained for . Thus it may be claimed that for resonant values of the resulting is not well-behaved enough (e.g. not smooth enough) for FURSTENBERG's theorem to apply. This interpretation is in agreement with the well-known fact that in the resonant case the quantum states typically (but not always) are delocalized [Fis93]. For nonresonant , and thus for a Lorentzian quasi-random distribution of the diagonal energies, it can be shown (see equation (B.6) in appendix B) that FURSTENBERG's integrability condition (B.2) is satisfied.

For the above reasoning the presence of the additional factor in the matrix element of equation (5.61) (as compared with equation (5.33)) is irrelevant: the condition (B.2) is satisfied for the matrix (5.61) if and only if it is satisfied for the matrix (5.33).

Note that the quantum resonances (5.67) are an entirely quantum mechanical phenomenon that does not have a classical counterpart. In this respect the quantum resonances of the kicked rotor are similar to the quantum resonances (4.7, 4.44, 4.49) of the resonant (with respect to ) kicked harmonic oscillator discussed in subsections 4.1.2 and 4.2.2. The resonances (1.23/4.22) with respect to , on the other hand, can also be viewed as a truly quantum mechanical phenomenon, but nevertheless they play an important role classically, too.

Putting all the above pieces together, a mapping from the quantum kicked
rotor
in the case of quantum nonresonance
onto the ANDERSON-LLOYD model
--
defined by equations
(5.33, 5.34, 5.41)
--
has been established, and
the results on
ANDERSON localization reviewed in the previous subsection carry
over
to the rotor. Using the same genericity assumptions as in subsection
5.1.2
(see pages f),
the averaged time-independent quasienergy
states
are found to be exponentially
localized
with respect to rotor (angular momentum) eigenstates;
the localization length
can be derived along the same lines as equation (5.42)
for , :

It is interesting to note the different roles the parameters and play with respect to the details of the localization mechanism. Regardless of the value of , the (resonant or nonresonant) value of alone decides on the existence of localization. , on the other hand, divided by controls the localization length via equations (5.59) and (5.71).

While by the above arguments it has been shown that there exists a
very
close relationship between the quantum kicked rotor and the ANDERSON model, it has to be stressed that until now no
mathematically *rigorous*
proof
has been found for exact equivalence of
both
model systems [CIS98], one of the reasons
being
that a tight binding equation for the rotor is obtained as an
-- albeit good --
approximation only;
in addition, the
consequences
of the quantum resonances (5.67)
for
such a potential proof are not yet
completely understood.
The ongoing effort to establish a complete analogy between the
quantum kicked rotor and an ANDERSON-like model is documented in
[AZ96,CIS98,AZ98],
for example.
The lack of mathematically rigorous equivalence may also be expressed by
noting that the
sample
kick potential (5.64), which
leads to the tight binding equation (5.66), differs
from the original kick potential (5.2).
Nevertheless, this explanation of localization in the quantum kicked rotor
is generally accepted, and all available numerical evidence supports the
equivalence of the two model systems, as far as localization is concerned
[Haa01].

In this way the numerical findings of subsection 5.1.1, especially the results on saturation of energy growth of the quantum kicked rotor displayed in figure 5.4, find a satisfactory explanation.

- ...
as
^{5.7} - In general, does not take on real values only. However, for kick potentials that are even with respect to , is real, thus justifying the characterization as an energy. The potential (5.2) falls into this category.
- ... matrix
^{5.8} - Without loss of generality can be assumed to be nonzero -- this assumption is identical with assuming nontriviality of the tight-binding equation (5.60) -- such that dividing by is not a problem: for the construction of the appropriate tight binding system it suffices to consider the first nonvanishing matrix element . A sample tight binding equation with is given below in equation (5.101). (But note that that equation is for the kicked harmonic oscillator rather than the kicked rotor; this accounts for some differences in details which are discussed there.)
- ...
quasi-random
^{5.9} - In the case of quantum-nonresonance in the sense of equation (5.67), despite the deterministic quadratic dependence on , effectively becomes randomly distributed in the interval by the implicit modulo operation, due to the periodicity of the tangent in equation (5.58).