<s>
The	B-Algorithm
kicked	I-Algorithm
rotator	I-Algorithm
,	O
also	O
spelled	O
as	O
kicked	B-Algorithm
rotor	I-Algorithm
,	O
is	O
a	O
paradigmatic	O
model	O
for	O
both	O
Hamiltonian	O
chaos	O
(	O
the	O
study	O
of	O
chaos	O
in	O
Hamiltonian	O
systems	O
)	O
and	O
quantum	O
chaos	O
.	O
</s>
<s>
The	O
equations	O
of	O
motion	O
of	O
the	B-Algorithm
kicked	I-Algorithm
rotator	I-Algorithm
writeTheses	O
equations	O
show	O
that	O
between	O
two	O
consecutive	O
kicks	O
,	O
the	O
rotator	O
simply	O
moves	O
freely	O
:	O
the	O
momentum	O
is	O
conserved	O
and	O
the	O
angular	O
position	O
growths	O
linearly	O
in	O
time	O
.	O
</s>
<s>
The	B-Algorithm
kicked	I-Algorithm
rotator	I-Algorithm
dynamics	O
can	O
thus	O
be	O
described	O
by	O
the	O
discrete	O
mapwhere	O
and	O
are	O
the	O
canonical	O
coordinates	O
at	O
time	O
,	O
just	O
before	O
the	O
-th	O
kick	O
.	O
</s>
<s>
It	O
is	O
usually	O
more	O
convenient	O
to	O
introduce	O
dimensionless	O
momentum	O
,	O
time	O
and	O
kicking	O
strength	O
to	O
reduce	O
the	O
dynamics	O
to	O
the	O
single	O
parameter	O
mapknown	O
as	O
Chirikov	B-Algorithm
standard	I-Algorithm
map	I-Algorithm
,	O
with	O
the	O
caveat	O
that	O
is	O
not	O
periodic	O
as	O
in	O
the	O
standard	B-Algorithm
map	I-Algorithm
.	O
</s>
<s>
However	O
,	O
one	O
can	O
directly	O
see	O
that	O
two	O
rotators	O
with	O
same	O
initial	O
angular	O
position	O
but	O
shifted	O
dimensionless	O
momentum	O
and	O
(	O
with	O
an	O
arbitrary	O
integer	O
)	O
will	O
have	O
the	O
same	O
exact	O
stroboscopic	O
dynamics	O
,	O
but	O
with	O
dimensionless	O
momentum	O
shifted	O
at	O
any	O
time	O
by	O
(	O
this	O
is	O
why	O
stroboscopic	O
phase	O
portraits	O
of	O
the	B-Algorithm
kicked	I-Algorithm
rotator	I-Algorithm
are	O
usually	O
displayed	O
in	O
a	O
single	O
momentum	O
cell	O
)	O
.	O
</s>
<s>
The	B-Algorithm
kicked	I-Algorithm
rotator	I-Algorithm
is	O
a	O
prototype	O
model	O
to	O
illustrate	O
the	O
transition	O
from	O
integrability	O
to	O
chaos	O
in	O
Hamiltonian	O
systems	O
and	O
in	O
particular	O
the	O
Kolmogorov	O
–	O
Arnold	O
–	O
Moser	O
theorem	O
.	O
</s>
<s>
More	O
precisely	O
,	O
after	O
kicks	O
,	O
the	O
momentum	O
of	O
a	O
particle	O
with	O
initial	O
momentum	O
writes	O
(	O
obtained	O
by	O
iterating	O
times	O
the	O
standard	B-Algorithm
map	I-Algorithm
)	O
.	O
</s>
<s>
After	O
a	O
careful	O
integration	O
of	O
the	O
time-dependent	O
Schrödinger	O
's	O
equation	O
,	O
one	O
finds	O
that	O
can	O
be	O
written	O
as	O
the	O
product	O
of	O
two	O
operatorsWe	O
recover	O
the	O
classical	O
interpretation	O
:	O
the	O
dynamics	O
of	O
the	O
quantum	O
kicked	B-Algorithm
rotor	I-Algorithm
between	O
two	O
kicks	O
is	O
the	O
succession	O
of	O
a	O
free	O
propagation	O
during	O
a	O
time	O
followed	O
by	O
a	O
short	O
kick	O
.	O
</s>
<s>
This	O
simple	O
expression	O
of	O
the	O
Floquet	O
operator	O
(	O
a	O
product	O
of	O
two	O
operators	O
,	O
one	O
diagonal	O
in	O
momentum	O
basis	O
,	O
the	O
other	O
one	O
diagonal	O
in	O
angular	O
position	O
basis	O
)	O
allows	O
to	O
easily	O
numerically	O
solve	O
the	O
evolution	O
of	O
a	O
given	O
wave	O
function	O
using	O
split-step	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
It	O
has	O
been	O
discovered	O
that	O
the	O
classical	O
diffusion	O
is	O
suppressed	O
in	O
the	O
quantum	O
kicked	B-Algorithm
rotator	I-Algorithm
.	O
</s>
<s>
The	O
quantum	O
kicked	B-Algorithm
rotor	I-Algorithm
can	O
actually	O
formally	O
be	O
related	O
to	O
the	O
Anderson	O
tight-binding	O
model	O
a	O
celebrated	O
Hamiltonian	O
that	O
describes	O
electrons	O
in	O
a	O
disordered	O
lattice	O
with	O
lattice	O
site	O
state	O
,	O
where	O
Anderson	O
localization	O
takes	O
place	O
(	O
in	O
one	O
dimension	O
)	O
where	O
the	O
are	O
random	O
on-site	O
energies	O
,	O
and	O
the	O
are	O
the	O
hoping	O
amplitudes	O
between	O
sites	O
and	O
.	O
</s>
<s>
In	O
the	O
quantum	O
kicked	B-Algorithm
rotator	I-Algorithm
it	O
can	O
be	O
shown	O
,	O
that	O
the	O
plane	O
wave	O
with	O
quantized	O
momentum	O
play	O
the	O
role	O
of	O
the	O
lattice	O
sites	O
states	O
.	O
</s>
<s>
The	O
full	O
mapping	O
to	O
the	O
Anderson	O
tight-binding	O
model	O
goes	O
as	O
follow	O
(	O
for	O
a	O
given	O
eigenstates	O
of	O
the	O
Floquet	O
operator	O
,	O
with	O
quasi-energy	O
)	O
Dynamical	O
localization	O
in	O
the	O
quantum	O
kicked	B-Algorithm
rotator	I-Algorithm
then	O
actually	O
takes	O
place	O
in	O
the	O
momentum	O
basis	O
.	O
</s>
<s>
Also	O
the	O
problem	O
of	O
quantum	O
kicked	B-Algorithm
rotator	I-Algorithm
with	O
dissipation	O
(	O
due	O
to	O
coupling	O
to	O
a	O
thermal	O
bath	O
)	O
has	O
been	O
considered	O
.	O
</s>
<s>
The	O
first	O
experimental	O
realizations	O
of	O
the	O
quantum	O
kicked	B-Algorithm
rotator	I-Algorithm
have	O
been	O
achieved	O
by	O
Mark	O
G	O
.	O
Raizen	O
group	O
in	O
1995	O
,	O
later	O
followed	O
by	O
the	O
Auckland	O
group	O
,	O
and	O
have	O
encouraged	O
a	O
renewed	O
interest	O
in	O
the	O
theoretical	O
analysis	O
.	O
</s>
