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\title{Quantum kicked rotors and related systems}
\author{Hiroshi FUJISAKI \thanks
{Institute of Physics, University of Tokyo,
%3-8-1 Komaba, Tokyo 153, Japan, 
e-mail: fujisaki@ASone.c.u-tokyo.ac.jp}}
%\date{}
\begin{document}
\maketitle
\section{Introduction}
A quantum kicked rotor (QKR) is one of the origins of {\it quantum chaos},
which roughly implies the field of study on {\it classically chaotic} quantum systems.
Since the first numerical investigation of a QKR by Casati {\it et al}. \cite{Casa},
there have appeared many descendants of it.
In order to settle the situation, I here review 
QKRs and the related systems which include a classical kicked rotor, 
an interacting boson system, an electron-phonon system and others.

\section{A classical kicked rotor (CKR)}

First of all, 
I discuss the properties of a {\it classical} kicked rotor (CKR).
The Hamiltonian for a CKR is,
\begin{eqnarray}
\label{HamCKR}
\mathfbox{
H(t) = p^2/2I +k V(\theta) \sum_n \delta(t-n),
}
\end{eqnarray}
where $p$ denotes an angular momentum, $I$ a moment of inertia, 
$K$ the strength of the nonlinearity,
and the period of the kick is normalized to be one.  
Note that this is a (periodically) {\it time-dependent} Hamiltonian,
and there is no {\it stochastic (noise) term} in it.
The equations of motion for a CKR {\it when $V(\theta)=\cos \theta$}, 
after some normalizations of the variables, are
\begin{eqnarray}
%\mathfbox{
\theta_{n+1} &=& (\theta_n +p_n) \hspace{2mm} {\rm mod} \hspace{1mm} 2 \pi,
\\
p_{n+1} &=& p_n+k \sin \theta_{n+1},
%}
\end{eqnarray}
where $n$ represents the discretized time step.
This is called {\it the standard map} \cite{Zasl,Ott}. 
Due to the delta function in Eq.\ (\ref{HamCKR}),
it {\it exactly} becomes a discrete map. 
So the numerical calculation is efficiently done.
Iterating the map, we can calculate, for example, the dynamics of the 
variance of $p$, and it was found that, under some conditions,
\begin{eqnarray}
\mathfbox{
\langle (p_n-p_0)^2 \rangle \simeq D n,
}
\end{eqnarray}
where the brackets mean the average for the initial ensemble,
and $D \simeq k^2/2$, which is numerically known.
This behavior is very similar to that of {\it diffusion}
[cf. $\langle (x_t-x_0)^2 \rangle \simeq D' t$], so 
this is called ``the diffusion of momentum".
This issue, which is understood as an example of the collapse of KAM tori and
the {\it spontaneous} emergence of stochasticity,
was intensively studied by Chirikov and his coworkers \cite{Chir}.

\section{A quantum kicked rotor (QKR)}

It is natural to ask what happens when one quantizes the CKR.
This was first numerically investigated by Casati {\it et al}. \cite{Casa}.
They employed the following 
Schr\"odinger equation for a {\it quantum} kicked rotor (QKR),
\begin{equation}
\label{QKR}
\mathfbox{
i \hbar \frac{\partial}{\partial t} \Psi(\theta,t)
= H(t) \Psi(\theta,t)
=\left[- \frac{\hbar^2}{2 I} \frac{\partial^2}{\partial \theta^2}
+k V(\theta) \sum_n \delta(t-n) \right] \Psi(\theta,t),
%= \{ H_0+\hat k V(\theta) \Delta(t)\} \Psi(\theta,t),
}
\end{equation}
where $H(t)$ is given by Eq.\ (\ref{HamCKR}) 
with the replacement $p=(\hbar/i) \partial/\partial \theta$.
Then, they numerically calculated the wavefunction of a QKR from 
Eq.\ (\ref{QKR}) as follows:
\begin{itemize}
\item
First, take the initial wave function. [e.g., $\Psi(\theta,t=0)=(2 \pi)^{-1/2}$.]
\item
Second, multiply the wave function by $\exp [-i(k/\hbar) V(\theta)]$.
\item
Fourier transform it [$\Psi(\theta) \rightarrow \Psi_l$].
\item
Multiply the Fourier transformed wave function by $\exp [-i \hbar l^2/2I]$.
\item
Inverse Fourier transform it [$\Psi_l \rightarrow \Psi(\theta)$].
\item
Return to the second step.
\end{itemize}
Iterating this loop, we obtain the dynamics of the wavefunction $\Psi(\theta,t)$,
and can calculate the quantum average of any physical quantity.
To discuss the relation between a CKR and a QKR, 
they calculated the variance of $p$ as follows,
\begin{equation}
\langle p_t^2 \rangle
=\hbar^2 \sum_{l=-\infty}^{\infty} l^2 |\Psi_l(t)|^2
\end{equation}
where the brackets mean the quantum average, and we have assumed, 
for simplicity, $\langle p_0 \rangle =0$, i.e., $\Psi(\theta,0)=(2 \pi)^{-1/2}$.
{\it When} $V(\theta)=\cos \theta$, they found, contrary to the case of a CKR, that
a QKR stops diffusing
\footnote{
As Izraelev and Shepelyansky showed \cite{IS},
for special values of $\hbar$,
there is the cases where $\langle p_t^2 \rangle \propto t^2$. 
This is called ``quantum resonance".}
in momentum space at a certain time $t^*$.
This result, {\it suppression of chaos in a quantum system}, seems to lead,
in general, to the absence of the {\it quantum stochasticity} \cite{Zasl,QS},
which denotes such concepts as mixing property  
or approaches to equilibrium states in quantum systems.
However, there are some ways to recover 
the stochasticity in quantum systems as discussed later.

\section{Mapping a QKR onto a tight-binding model}

Mapping is a useful concept when we understand something deeper.
In this section, following Fishman {\it et al}. \cite{Fish}, we map {\it one} QKR
onto another problem, which is related to 1D Anderson Localization (AL) \cite{Ishi}.
Since the issue on AL has been well studied in solid-state physics \cite{Lee},
we can, to understand QKRs, use the method developed in the AL literature,
and vice versa.

In the following, I review the procedure in detail:
The Hamiltonian in Eq.\ (\ref{QKR}) has the periodicity, $H(t+1)=H(t)$,
so the Floquet's theorem can be applied, i.e., the wavefunction can be written as 
$\Psi(\theta,t)=e^{-i \omega t}u_\omega (\theta,t)$, where
$u_\omega(\theta,t+1)=u_\omega(\theta,t)$ 
and $\omega$ is called a {\it quasi-energy} (which is like a Bloch index
in the field of solid-state physics \cite{Kitt}).
Hence, all we need is the dynamics from
$t=0_+$ (just after the kick at $t=0$) to $t=1$.
Since the particle freely moves between $t=0_+$ and $t=1_-$
(just before the kick at $t=1$),
the transformation is 
\begin{equation}
\label{free}
u_{-,m}=u_{+,m}e^{i E_m},
\end{equation}
where
$u_+(\theta)=u_+(\theta, t=0_+)$, $u_-(\theta)=u_-(\theta, t=1_-)$,
$E_m=\omega -\tau m^2/2$, $\tau=1/I$, and 
we have defined the Fourier component of $f(\theta)$, $f_m$, as:
\[
f(\theta) = \sum_m f_m e^{i m \theta}, \hspace{5mm}
f_m = \frac{1}{2\pi}\int_0^{2\pi} d\theta f(\theta) e^{-i m \theta}.
\]
Inverse Fourier transforming Eq.\ (\ref{free}), we obtain
\begin{equation}
\label{free2}
u_-(\theta)=\sum_m \int \frac{d \phi}{2 \pi} e^{i m(\theta-\phi)+i E_m} u_+(\phi).
\end{equation}
Between $t=1_-$ and $t=1_+=0_+$, 
the particle feels a kick and is transformed to be 
\begin{equation}
\label{kick}
u_+(\theta)=u_-(\theta) e^{-i k V(\theta)}.
\end{equation}
Next, we define $\bar u = (u_+ +u_-)/2$, and have, from Eq.\ (\ref{kick}),
$\bar u = u_+/(1-i U)=u_-/(1+i U)$, where $U=-\tan[k V(\theta)/2]$.
Substituting these into Eq.\ (\ref{free2}) to eliminate $u_+, u_-$,
we have
\begin{eqnarray}
[1-i U(\theta)] \bar u(\theta)
&=&
\sum_m \int \frac{d \phi}{2\pi} e^{i m(\theta-\phi)+i E_m}
[1+i U(\phi)] \bar u(\phi)
\nonumber
\\
&=&
\sum_m e^{i (E_m +m \theta)} \int \frac{d \phi}{2\pi}
\left[ 
\sum_r u_r e^{i(r-m) \phi}+i \sum_n \sum_r U_n u_r e^{i(n+r-m) \phi} 
\right]
\nonumber
\\
&=&
\sum_m e^{i (E_m +m \theta)}
\left[
u_m +i \sum_r U_{m-r} u_r 
\right].
\nonumber
\end{eqnarray}
Fourier transforming the above equation, 
we obtain the left-hand side (l.h.s.) and right-hand side (r.h.s.) of the equation as
\begin{eqnarray}
{\rm l.h.s.}
&=&
u_n -i \sum_r U_{n-r} u_r,
\nonumber
\\
{\rm r.h.s.}
&=&
e^{iE_n} (u_n +i \sum_r U_{n-r} u_r).
\nonumber
\end{eqnarray}
Equating these, we finally have
\begin{equation}
\label{AL}
%\tan(E_n/2) u_n +\sum_r U_{n-r} u_r =
\mathfbox{
T_n u_n +\sum_r U_{n-r} u_r =0,
}
\end{equation}
where $T_n =\tan(E_n/2)$.
This is just the tight-binding model in solid-state physics \cite{Kitt},
if we regard the subscript ``r" of $u_r$ as 
a ``site" in {\it momentum} space,
$T_n$ as a site energy, and $U_r$ as a hopping energy.
In the AL literature, the site energy $T_n$ is a stochastic variable,
whereas it is just a {\it deterministic function} of $n$ here.
In spite of this fact, if $\tau$ is an {\it irrational number},
$T_n =\tan(E_n/2)$ behaves {\it like random},
\footnote{
This is also called {\it quasi-random} (or {\it pseudo-random}).
In fact, since, in numerical calculations, 
some algorithms must be used to generate random numbers,
{\it only} quasi-random numbers are available for us.
As Kolmogorov, Chaitin, and Solomonoff independently showed,
we, {\it in  principle}, can {\it not} judge which number is {\it exactly} random
though we {\it can} define randomness \cite{Chai}.
This fact is,  as the halting problem of a (universal) Turing machine \cite{Penr},
another version of G\"odel's incompleteness theorem \cite{Gell}.}
and AL {\it does} appear even if Eq.\ (\ref{AL}) is employed!
This is a very interesting and striking result 
that Fishman {\it et al}. showed in their numerical calculations.

On the other hand, if $\tau$ is a rational number,
$T_n$ becomes a periodic function of $n$.
Hence, all the states become extended (Bloch) ones, 
in other words, AL disappears.
They considered that this case is rather rare in physics, 
and that AL always will appear.
However, because they did not find an {\it exact} link between a QKR and 1D AL,
their argument, in a sense, was not a complete one.
Recently, Altland and Zirnbauer \cite{AZ} showed that the behaviors of a QKR 
{\it in low-frequency regions} can be mapped onto 
a supersymmetric nonlinear sigma (SUSYNS) model.
On the other hand, it is known that AL problem {\it without
electron-electron interactions} also can be mapped onto the same model \cite{Efet}.
Therefore, the relation, 1D AL $\sim$ a SUSYNS model $\sim$ a QKR was established.
Strictly speaking, however, their argument only concerned with 
a link between {\it one} QKR and 1D AL. 
What happens when we extend one-body QKR to 
many-body (or higher dimensional) ones ?

\section{Extension to many-body QKR systems}

There are many ways to extend a QKR to many-body ones.
One possibility is the following Hamiltonian,
\begin{equation}
\label{HamMB}
\mathfbox{
H(t)=
- \frac{\hbar^2}{2 I} \sum_i \frac{\partial^2}{\partial \theta_i^2}
+k V(\{\theta_k \}) \sum_n \delta(t-n),
}
\end{equation}
where $V(\{\theta_k \})=V(\theta_1, \theta_2,..., \theta_N)$ can be complex.
In particular, the following form of the potential energy,
\begin{eqnarray}
\mathfbox{
V(\{\theta_k \}) 
%&=& \sum_i V^{(1)}(\theta_i) + \sum_{i \neq j} V^{(2)}(\theta_i - \theta_j)
%\nonumber
%\\
= \sum_{i=1}^N K_i \cos \theta_i +
\sum_{i \neq j}^N \epsilon_{ij} \cos (\theta_i-\theta_j),
}
\end{eqnarray}
was examined by Toda, Adachi, and Ikeda \cite{TAI}. 
In a physical sense, this is a natural choice because it resembles
a Heisenberg XY model in the field of magnetism.
They showed in their numerical calculations that
even if $N$ (the total number of QKRs) $=2$, there appears, in a sense, 
again the diffusion of the momentum which exists in a CKR. 

Now let us map this ($N=2$-QKR) problem onto 2D AL.
This is a generalization of the method by Fishman {\it et al.}:
\begin{equation}
%\label{AL}
%\tan(E_n/2) u_n +\sum_r U_{n-r} u_r =
T_{\bf n} u_{\bf n} 
+\sum_{\bf r} U_{\bf n-r} u_{\bf r} =0,
\end{equation}
where ${\bf n} =(n, m)$ is a {\it pair} of integers.
If we use the analogy with 2D AL, there should be 
localization due to the scaling argument \cite{Four}.
According to Toda {\it et al.}, however, {\it there is not !}
This remarkable discrepancy, I think, deserves further considerations.
{\bf We need to calculate the localization length 
and/or something else in 2D (or higher) momentum space},
like Shepelyansky did in the 1D space \cite{Shep}.
We also must investigate how the property of the hopping integral
$U_{\bf n-r}$ affects the result. In the textbook, it is often assumed
that $U_{\bf n-r}$ has only nearest-neighbor components,
but it is not the case in general.
\footnote{
Of course, after renormalization procedure, there might appear a universality
where {\it only} nearest-neighbor hopping is relevant.}

Furthermore, it is known that a 3D-random system exhibits 
the metal-insulator (MI) transition. 
So, {\bf what is the correspondence of the MI transition
in quantum chaos ?} %\cite{Bran}
This issue was considered by Casati {\it et al}. \cite{CGS}
in the following many-body QKR model,
\begin{equation}
\label{Casa}
H(t)=H_0(n)+\omega_1 n_1 +\omega_2 n_2 
+V(\theta, \theta_1, \theta_2) \sum_n \delta(t-n),
\end{equation}
where $H_0(n)$ is a random Hamiltonian (this might be a trick !),
$n_i=-i\partial/\partial \theta_i$ ($i$=1,2), and $\omega_1, \omega_2$ 
incommensurate frequencies. 
There are three degrees of freedom, $\theta, \theta_1, \theta_2$, 
so we can map this onto 3D AL. They, in particular, took the following potential,
\[
V(\theta, \theta_1, \theta_2)
=-2 \tan^{-1} [2 k(\cos \theta+\cos \theta_1+\cos \theta_2)],
\]
for the mapped model to have {\it only} nearest-neighbor hopping.
%\[
%T_{\bf n} u_{\bf n} +k {\sum_{\bf n}}' u_{\bf n}=0,
%\]
%where $T_{\bf n}=-\tan[\frac{1}{2}(E_n+n_1 \omega_1 +n_2 \omega_2+\lambda)]$
%($\lambda$ is quasienergy, $E_n$ random energy distributed in $[0, 2 \pi]$)
%and the sum $\sum'$ includes only nearest-neighbor hopping,
%is easier to calculate. 
They then numerically calculated the localization length in the case of localization,
and the diffusion of momentum in the case of delocalization, and
showed that there occurs the localization-delocalization transition corresponding to 
the MI transition in 3D-random systems. 

There exist other generalizations of a QKR \cite{Shep2,Iked}.
Since the number of the generalizations is, of course, enormous,
{\bf we need some phenomenology to classify many-body QKR systems} 
for economic reasons. This issue remains to be investigated.

\section{Many-body quantum systems except QKRs}

Next, we will consider some many-body systems except QKRs.
The first example is the {\it quantum hard sphere gas} \cite{Gasp}, 
which is described by the following Schr\"odinger equation,
\begin{equation}
-\frac{\hbar^2}{2 m} \sum_a \nabla_a^2 \Psi =E \Psi,
\end{equation}
where $\Psi=0$ at the boundary defined by
%\begin{eqnarray}
\[
| {\bf r}_a -{\bf r}_b | \geq d,
\hspace{2mm}
d/2 \leq x_a, y_a, z_a \leq L-d/2.
\]
%\end{eqnarray}
Its classical counterpart was studied by Krylov and Sinai \cite{Kryl},
who showed that the classical hard sphere gas has positive 
{\it Kolmogorov-Sinai entropy}, 
which is one of the indicators of stochasticity in classical physics. 
For the semiclassical treatment of quantum hard sphere gas,
see Ref. \cite{Gasp}. 

The second is the interacting boson system studied by 
Jona-Lasinio and Presilla \cite{Jona}.
Their Hamiltonian is 
\begin{equation}
\label{Jona}
H=\sum_{j=1}^L [\alpha_j a_j^{\dagger} a_j
-\beta_j(e^{i \theta} a_{j+1}^{\dagger} a_j +{\rm h.c.})]
+\sum_{j=1}^L \gamma_j a_j^{\dagger} a_j^{\dagger}a_j a_j,
\end{equation}
where the operator $a^{\dagger}_j$ creates a boson in the site $j$,
and $\alpha_j, \beta_j$ and $\gamma_j$ are the site, hopping, and interaction 
energies, respectively.
They numerically calculated the following quantity,
\begin{equation}
\kappa_{AB}=\lim_{t \rightarrow \infty}
\sqrt{\frac{1}{t}\int_0^t dt'
|\langle A(t) B \rangle - \langle A \rangle \langle B \rangle|},
\end{equation}
which indicates the potential of a mixing property, i.e., the maximal mixing implies
that $\kappa_{AB}=0$.
They showed that as the number of the particles increases 
(although the calculated number is only up to seven),
$\kappa_{AB}$ decreases to zero. This result agrees with our intuition.
They also studied its classical counterpart with mean-field approximation.
The approximated equation of motion derived from Eq.\ (\ref{Jona}) is
\begin{equation}
\label{Jona2}
i \hbar \frac{d}{dt} z_j(t)
=[\alpha_j +2(N-1) \gamma_j |z_j(t)|^2] z_j(t)
-\beta_{j-1} e^{i \theta} z_{j-1}(t) 
-\beta_{j} e^{-i \theta} z_{j+1}(t),
\end{equation}
where $z_j(t)$ is the amplitude of the mean field in the site $j$.
They numerically showed that when the classical system described 
by Eq.\ (\ref{Jona2}) becomes chaotic {\it in a classical sense},
the corresponding quantum system described by Eq.\ (\ref{Jona}) has a mixing property.
This is one of the examples of {\it quantum-classical correspondence} 
studied in the field of quantum chaos.
Similar {\it spin} systems were also treated by Jensen, Shankar
and others \cite{JS}, and it was shown that even {\it closed small}
integrable (and, of course, non-integrable) systems {\it numerically} exhibit 
a mixing property.

The third is the 1D-random electron system coupled with {\it a few modes} 
of phonon,
which was studied by Yamada and Ikeda \cite{YI}. Their Hamiltonian is 
\begin{eqnarray}
\label{YI}
H &=&H_{\rm el}+H_{\rm ph}+H_{\rm int},
\nonumber
\\
&=&\sum_{n=1}^N |n \rangle V(n) \langle n | 
+ \sum_{n \neq m}^N |n \rangle K(n,m) \langle m |
\nonumber
\\
&&+\sum_{j=1}^M (\frac{1}{2} \hat p_j^2 +\frac{1}{2} \omega_j^2 \hat q_j^2)
+\sum_{n=1}^N \sum_{j=1}^M  [|n \rangle V(n) \langle n |]b_j \hat q_j,
\end{eqnarray}
where $V(n)$ is the random site energy distributed in $[-W,W]$,
 $K(n,m)$ the hopping energy,
$\hat q_j, \hat p_j$ the coordinate and momentum of phonons, respectively,
$b_j$ the coupling strength between the electron and phonons.
(A similar, but less quantum, model was examined 
by Kopidakis {\it et al}. \cite{KSE}. 
One may point out the similarity between Eq.\ (\ref{Casa}) and above.)
They numerically calculated from Eq.\ (\ref{YI}) 
the mean-square displacement, 
\[
\langle \Delta \hat n^2 \rangle 
= \langle \Psi(t)| (\hat n - \langle \hat n \rangle)^2 | \Psi(t) \rangle,
\]
where $\hat n=\sum_{j=1}^N j |j \rangle \langle j|$ is the position operator
and $| \Psi(t) \rangle$ is the time-dependent wave function.
This quantity characterizes whether the electron becomes localized or extended.
Their numerical experiments showed that there is an anomalous diffusion law,
$\langle \Delta \hat n^2 \rangle =D t^{\alpha}$,
and $\alpha$ approaches one from below 
as the number of the phonon mode increases.
(When the number is {\it three}, we can conceive normal diffusion:
$\langle \Delta \hat n^2 \rangle \simeq D t$.)
At the same time, they calculated the energy of the electron system,
$E_{\rm el}(t)=\langle \Psi(t)| H_{\rm el}| \Psi(t) \rangle$,
and that of the phonon system, 
$E_{\rm ph}(t)=\langle \Psi(t)| H_{\rm ph}| \Psi(t) \rangle$.
They showed, when the coupling $b_j$ 
and the randomness $W$ are strong enough, there exists 
``a one-way quasi-stationary energy flow from electron to phonons" \cite{YI}.

Next, we see, in Yamada-Ikeda model, what happens when all the phonon modes 
are excited in coherent states, i.e., all the phonons are {\it classicalized}.
\footnote{
They instead considered the case when all the phonon modes 
are excited around {\it Fock state} with a large quantum number \cite{YI}.
But, this, I think, is a rather unphysical choice, because 
the states are highly nonclassical.}
%Then, using the same procedure 
%in the field of quantum optics \cite{Loud}, we can map Eq.\ (\ref{YI})
%onto the time-dependent Hamiltonian which does not contain 
%the degrees of freedom of the phonons.
The procedure is as follows:
Unitary transform the Schr\"odinger equation 
$i \hbar \frac{\partial}{\partial t} \Psi(t)
= (H_{\rm el}+H_{\rm ph}+H_{\rm int}) \Psi(t)$
into another one for $\phi(t)=\exp(i H_{\rm ph} t/\hbar) \Psi(t)$,
which satisfies,
\[
i \hbar \frac{\partial}{\partial t} \phi(t)
= [H_{\rm el}+ 
e^{i H_{\rm ph} t/\hbar} H_{\rm int} e^{-i H_{\rm ph} t/\hbar}]\phi(t).
\]
We can expand $e^{i H_{\rm ph} t/\hbar} H_{\rm int} e^{-i H_{\rm ph} t/\hbar}$ as
\begin{eqnarray}
e^{i H_{\rm ph} t/\hbar} H_{\rm int}e^{-i H_{\rm ph} t/\hbar}
&=&\sum_{n=1}^N \sum_{j=1}^M  [|n \rangle V(n) \langle n |]b_j 
\sqrt{\frac{\hbar}{2 \omega_j}}(\hat a_j e^{-i \omega_j t} +{\rm h.c.})
\nonumber
\\
&\simeq& 
\sum_{n=1}^N \sum_{j=1}^M  [|n \rangle V(n) \langle n |]
\epsilon_j \cos(\omega_j t+\phi_j),
\end{eqnarray}
where
\begin{eqnarray}
\epsilon_j &=&
\sqrt{2 \hbar b_j^2 |\alpha_j|^2/\omega_j},
\hspace{2mm}
\hat q_j = \sqrt{\hbar/2 \omega_j}(\hat a_j+\hat a_j^{\dagger}),
\nonumber
\\
H_{\rm ph}
&=& \sum_{j=1}^M \hbar \omega_j (\hat a_j^{\dagger}\hat a_j+1/2),
\hspace{2mm}
e^{i H_{\rm ph} t/\hbar} \hat a_j e^{-i H_{\rm ph} t/\hbar}
= \hat a_j e^{-i \omega_j t},
\nonumber
\end{eqnarray}
and we have used the following {\it approximation},
\[
\hat a_j \simeq  |\alpha_j| e^{-i \phi_j},
\]
(where $\alpha_j, \phi_j$ are c-numbers)
which is justified when all the phonons are {\it classicalized}
due to decoherence or something.
Hence, the reduced system is described by
\[
H'_{\rm el}
=\sum_{n=1}^N |n \rangle V(n,t) \langle n | 
+ \sum_{n \neq m}^N |n \rangle K(n,m) \langle m |,
\]
where
\[
V(n,t)=V(n)[1+\sum_{j=1}^M \epsilon_j \cos(\omega_j t+\phi_j)].
\]
This system, with incommensurate frequencies $\omega_j$, 
was also investigated by them and Goda \cite{YIG}.
It was shown that the periodic perturbation causes the destruction of 
localization.
%which is similar to the behaviors of Harper's model,
%\[
%H_{\rm Harper}
%=\sum_{n=1}^N |n \rangle \nu(n) \langle n | 
%+ \sum_{n \neq m}^N |n \rangle K(n,m) \langle m |,
%\]
%where 
%$\nu(n)=\nu_0 \cos(2 \pi g n+\eta)$, $g$ is an irrational number, and 
%$\eta$ a phase. (An atom-optics realization of this model was discussed 
%in Ref.\ \cite{DH}.) 
A little different system in {\it more than two dimensions}
is being examined by Brandes \cite{Bran}.

The systems discussed above are only a few examples of 
many-body quantum chaos systems.
There will be, of course, other interesting systems to study 
the quantum stochasticity.


\section{Experimental aspects of quantum chaos systems}

In this section, I discuss the experimental aspects of quantum chaos systems.
%which is, e.g., observed in a QKR up to a certain time $t^*$ as discussed above.
From a historical point of view, 
the results of experiments \cite{Bayf} on ionization of Rydberg atoms by an
external field were first shown to have the properties of a quantum chaos
system \cite{CCSG}.
The Hamiltonian considered is
\begin{equation}
H_0(t)=\frac{p^2}{2m}-\frac{e^2}{4 \pi \epsilon_0} \frac{1}{x}
+e x F \sin(\omega t), \hspace{2mm} x>0,
\end{equation}
where $F$ and $\omega$ are the field strength and frequency in atomic units.
This is a model rather different from a QKR, but it has similar consequences,
including quantum limitation of chaotic excitation (ionization). 
In real situations, of course, {\it noise} affects the behaviors of the system considered.
This issue was investigated by Bl\"umel {\it et al}. \cite{Blum}.
Their total Hamiltonian, including the variables for the environment, is 
\[
H(t)=H_0(t)+H_{\rm int}+H_{\rm R},
\]
with 
%\begin{eqnarray}
\[
H_{\rm int}=\hbar \sum_i g(\omega_i) x (b_i +b_i^{\dagger}),
\hspace{2mm}
H_{\rm R}=\hbar \sum_i \omega_i (b_i^{\dagger}b_i+1/2).
\]
%\end{eqnarray}
Their central result is that small amount of noise causes the destruction of 
quantum coherence and the localization in energy space, which is the origin of 
quantum limitation of chaotic excitation, disappears.
This is very similar to the results in the case of a QKR \cite{OA}.
In solid-state physics literature, 
this is similar to those by Thouless \cite{Thou} who considered the effect of 
finite temperature on localization in a solid.

Those discussed above are not directly related to QKRs.
But, in the field of atom optics, there exists an ideal realization of a QKR \cite{Moor}.
The situation is as follows:
An atom is exposed to the delta-pulse trains of light,
which simultaneously simulate the potential and the delta functions in Eq.\ (\ref{QKR}). 
It was shown that the atom-optics system of a QKR
{\it experimentally} exhibits 
the same behavior as predicted by Casati {\it et al}. \cite{Casa}.

Atom-optics systems are now being studied 
in connection with Bose-Einstein condensation. %\cite{BEC}.
I think {\bf these systems are also promising, 
both experimentally and theoretically, 
to investigate quantum chaos sytems, including many-body QKR ones, 
and to do quantum stochasticity}.


\section{Summary}
Quantum kicked rotors (QKRs) are typical examples in the field of quantum chaos.
A QKR causes the suppression of chaos in very short time scale,
and that is all. However, if there are more degrees of freedom, many things can happen.
We need more calculations for many-body QKR systems,
and some phenomenology to classify them.
In order to achieve these, it is important 
to map them onto other systems (including tight-binding models,
a supersymmetric nonlinear sigma model, and many others),
because the crossreference which occurs there makes our understanding deeper.
Atom-optics systems seem to be experimentally best
to test the theoretical calculations on QKRs.
Of course, except QKRs, there are many other quantum chaos systems,
and many things remain to be done.


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\end{thebibliography}
\end{document}
