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\textbf{
Additional authors for the experimental section:
}
Daniel A. Steck,
Windell H. Oskay, and
Mark G. Raizen
(all affiliated with UT-Austin)
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\textit{
Text to be tacked onto the end of the abstract:
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These results are also applied to experimental measurements
in the quantum kicked rotor system.
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% Experimental Figure here
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\begin{figure}[h]
\begin{center}
\includegraphics*[trim=0in 0in 0in 0in, scale=0.48]
{composite_plot.eps}
\end{center}
\caption{Experimental quantum kicked rotor data (heavy lines)
with best fits (open circles), shown only at the
time of 70 kicks for clarity. The three regimes shown
are (a) exponential localization, (b) anomalous
diffusion, and (c) noise-induced delocalization, for
which the respective exponents are $\gamma = 1.06$,
1.48, and 2.03.
\label{expt_fig}}
\end{figure}
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We also used Eqs.~(1) and (2) to analyze experimental momentum
distributions of the quantum kicked rotor. The details of the
atom-optics experiment and the measurement conditions
are described in Ref.~\cite{Steck00}.
In each case that we study, the distributions after 30, 40, 50,
60, and 70 kicks are fit simultaneously to Eq.~(1) while imposing
Eq.~(2) as a constraint. To more accurately reflect the experiment,
the fundamental distribution in Eq.~(2) is convolved with the
initial atomic momentum distribution, and a correction is applied
for the known response of the detection system.
In order to obtain more accurate fits, the center of the
distribution ($|p/2\hbar k_{\mathrm{L}}|<15$, where
$\hbar k_{\mathrm{L}}$ is the photon recoil momentum) is excluded
from the fit, along with the extreme tails
($|p/2\hbar k_{\mathrm{L}}|>70$), where the signal levels are
unreliably small and where ``momentum boundary'' effects
are most significant \cite{Steck00}.
We study three distinct cases, beginning with dynamical localization
(with a stochasticity parameter $K=11.2$),
where we find an exponent $\gamma = 1.06 \pm 0.19$,
which is consistent with exponential localization.
In the next case, the
corresponding classical dynamics exhibit
anomalous diffusion ($K=8.4$), where the fit yields
$\gamma = 1.48 \pm 0.16$; this larger exponent is consistent
with the curved, nonexponential distribution tails observed
in the experiments \cite{Klappauf98}.
Finally, we study the kicked rotor with $K=11.2$ driven strongly
(200\%) by amplitude noise, where the dynamics
mimic classical diffusion; in this case we find
$\gamma = 2.03 \pm 0.14$, which is consistent with
normal diffusion.
The largest contribution to the quoted uncertainties is the
sensitivity of the fit to changing the cutoff boundaries
described above.
The data and fits for all three cases
are shown at 70 kicks in Fig.~\ref{expt_fig}, showing excellent
agreement in the tails of the distributions.
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\textit{
Additions to the acknowledgements section
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NSF grant no.~PHY-9987706.
D.A.S. acknowledges support from the Fannie and John Hertz
Foundation.
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\begin{thebibliography}{99}
\bibitem{Steck00}
D.~A. Steck, V. Milner, W.~H. Oskay, and M.~G. Raizen,
\pre \textbf{62}, 3461 (2000).
\bibitem{Klappauf98}
B.~G. Klappauf, W.~H. Oskay, D.~A. Steck, and M.~G. Raizen,
\prl \textbf{81}, 4044 (1998).
\end{thebibliography}
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