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Quantum Chaos and Atom Optics:
from Experiments to Number Theory
Italo Guarneri, Laura Rebuzzini, Michael Sheinman
Sandro Wimberger, Roberto Artuso and S.F.
Experiments: M. d’Arcy, G. Summy, M. Oberthaler, R. Godun, Z.Y. Ma
Collaborators: K. Burnett, A. Buchleitner, S.A. Gardiner, T. Oliker, M.
Sheinman, R. Hihinishvili, A. Iomin
,
Advice and comments: M.V. Berry, Y. Gefen, M. Raizen, W. Phillips, D.
Ullmo, P.Schlagheck, E. Narimanov
Quantum Chaos
Kicked Rotor
Classical
Quantum
Atom Optics
Diffusion (1979 )
Deviations from classical behavior
Anderson localization (1958,1982)
Fictitious
Classical mechanics
Far from the classical
limit (2002)
Observation of Anderson localization for
laser cooled Cs atoms (Raizen, 1995)
Quantum nonlinear
resonance
Short wavelength perturbation
Effects of gravity, Oxford 1999
New resonance
Experiment
R.M. Godun, M.B.d’Arcy, M.K. Oberthaler, G.S. Summy and K. Burnett, Phys. Rev. A 62, 013411 (2000),
Phys. Rev. Lett. 83, 4447 (1999)
Related experiments by M. Raizen and coworkers
1. Laser cooling of Cs Atoms
2. Driving
e
g
L
d E
E
Electric field
potential
x
E  d  E 2
V  cos Gx (t  mT )
m
Mgx
3. Detection of momentum distribution
On center of mass
dipole
Experimental results
relative to free fall
any
structure?
 / 2
1  67 s
p=momentum
Accelerator mode
What is this mode?
Why is it stable?
What is the decay mechanism and the decay rate?
Any other modes of this type?
How general??
Kicked Rotor Model
F

F
H 
2
2I
nˆ 2  K cos   (t  mT )
m

nˆ  i

Dimensionless units
k
K
 
T
I
1 2
H =  nˆ  k cos   (t  m)
2
m
( p n K   k)
Classical Motion
m
m
 m1
pm m  1
 m1   m  pm
(0 , p0 )
Assume
p0  0
K  2
kick
  /2
p1  2
p2  4
pm  2 m
Accelerated , also vicinity accelerated
Robust , holds also for vicinity of
p
t
Standard Map
pm1  pm  K sin  m1
 0   / 2 2
1   / 2  2
 2   / 2  4
   / 2  2 m
pm1
K  2

 p2 
t
For typical
K
1
sin m
kick
Effectively random
kick
kick
 p2 
Diffusion in
t0
kick
t
For values of
Nonlinearity
K
Where acceleration , it dominates
Accelerator modes robust
p
t0
p
( p n K   k)
Classical Motion
 t 1   t  pt
Standard Map
pt 1  pt  K sin  t 1
For typical
K
1
sin  t
Effectively random
 p2 
Diffusion in
t0
Diffusion
p
t0
t
for
some
K  2  integer
(0 , p0 )
for example
and vicinity accelerated
( / 2, 0)
Acceleration
 p2 
t
1 2
H =  nˆ  k cos   (t  m)
2
m
Quantum
Uˆ t   t 1
Evolution operator
Anderson localization

e


2
 i nˆ 2
2
rational



2
Uˆ  e
T

 

I 


 i nˆ 2
2
e
 ik cos
irrational
pseudorandom
Quantum resonance
Anderson localization like
for 1D solids with disorder
 p2  t 2
1 2
H =  nˆ  k cos   (t  m)
2
m
Quantum
 / 2 

irrational
e
 i nˆ 2
2

T

 

I 

pseudorandom
Anderson localization like
for 1D solids with disorder
 n2 
classical
Eigenstates of
Uˆ
Exponentially localized
quantum
t
 / 2  rational
Simple resonances:
4 
Quantum resonance
 p2 
  2 , 4 ...2 l
Talbot time
t
Kicked Particle
V ( x)

1 2
H =  pˆ  k cos x  (t  m)
2
m
rotor
Quantum :
cos x
  x    p
periodic

Not quantized
transitions
p pn
Classical-similar to rotor
  fractional part of p (quasimomentum ) CONSERVED
 / 2  rational, resonance only for few values of 
 / 2  irrational
Anderson localization
2
p 
P( p)
t
classical
p
quantum
t
1 2
H =  pˆ  k cos x  (t  m)
2
m
kicked rotor
0    x  2
classical
K  k
typical K
K  2 l
diffusion in
  typical
 / 2  rational
  x  
p
diffusion in
acceleration
p  integer
quantum
kicked particle
Localization in
p
acceleration
p
arbitrary
p pn
p
resonances
Localization in
p
resonances only for few
initial conditions
1
(momentum) 2
2
k2
t
2

2
<
t
momentum
t
F.L. Moore, J.C. Robinson,
C.F. Bharucha, B. Sundaram
and M.G. Raizen, PRL 75,
4598 (1995)
Effect of Gravity on Kicked Atoms
Quantum accelerator modes
A short wavelength perturbation superimposed on long wavelength behavior
Experiment-kicked atoms in presence of gravity
p2

H 
 Mgx  1  cos Gx    (t  mT )
2M
2
m
G  4 / 
  895nm
Gx  x
dimensionless units
T  66.5 s  l
t /T  t
H 
1 2
 H =  pˆ   x  k cos x  (t  m)
2
m
 TG 2 
 

M


in experiment
x

2
 MT 
 
g
G 
k 
  0.1
k
NOT periodic
quasimomentum NOT conserved
x
NOT periodic
quasimomentum NOT conserved
gauge transformation to restore periodicity
  2 l  
l

integer
introduce fictitious classical limit where

1
plays the role of
Gauge Transformation
1 2
H I =  pˆ   x  k cos x  (t  m)
2
m
1
2
H II =   pˆ  t   k cos x  (t  m)
2
m

i   H I
t

i   H II 
t
 ( x, t )  e
same classical equation
for x
 ( x, t )
i xt
H II momentum relative to free fall ( t )
quasimomentum 
conserved
p  n

nˆ  i
  x mod(2 )

For
Uˆ  Uˆ kick  Uˆ free
Quantum Evolution
ik cos
ˆ
Ukick  e
Uˆ free  e
Uˆ free  e
Uˆ free  e
1

 i   nˆ 2  nˆ   t  / 2  
2

  2 l  
up to terms independent of
operators but depending on
i n2l
e
 1 ˆ2 ˆ
 i   n  n l  nˆ   t  / 2  
2

i 

  Iˆ 2  Iˆ l  (  t  / 2)  
| |  2

Uˆ kick  e
“momentum”
i
k
| |
cos
i nl
e
  sign( )

Iˆ |  | nˆ  i |  |

k  k | |

ˆI  i |  |  “momentum”

Uˆ  e
i
k
cos
| |

e
i
| |
k  k | |
destroys localization
H
1 ˆ2 ˆ
H    I  I  l   (   t   / 2) 
2
dynamics of a kicked system where
|  | plays the role of
|  |effective Planck’s constant
quantization

p  i
x
Fictitious classical mechanics useful for
meaningful “classical limit”
dequantization i |

| I

|  | 1 near resonance
 -classical dynamics
H = H   k cos  (t  m)
m
It 1  It  k sint 1
change variables
t 1  t  It  t   l     / 2
Jt  It  t   l    / 2
Jt 1  Jt  k sint 1 
motion on torus
   mod(2 )
t 1  t  Jt
J = J mod(2 )
Accelerator modes
Jt 1  Jt  k sint 1 
motion on torus
t 1  t  Jt
J = J mod(2 )
   mod(2 )
Solve for stable classical periodic orbits
follow wave packets in islands of stability
stable
quantum accelerator mode
period 1 (fixed points):
J0  0
solution requires choice of

and

-classical periodic orbit
sin0   / k
0
accelerator mode
n  n0  t / 

Color --- Husimi (coarse grained Wigner)
black
-classics

Color-quantum
Lines
classical
Experimental results
relative to free fall
any
structure?
 / 2
1  67 s
p=momentum
Accelerator mode
What is this mode?
Why is it stable?
What is the decay mechanism and the decay rate?
Any other modes of this type?
How general??

Color-quantum
Lines
classical
P e

e
 A/
  t
e
decay mode
 A /| |
transient
decay rate
Accelerator mode spectroscopy
map:
Jt 1  Jt  k sint 1 
period
motion on torus
n  I / | |
p
fixed point
t 1  t  Jt
J p  J 0  2 j
 p   0  2 n
2 
| j |
n  n0  t



| | 
p
Higher accelerator modes:

difference from
rational
( p, j)  (period, jump in momentum)
observed in experiments
j/ p
Acceleration
proportional to
as Farey approximants of
gravity in some units


mod(1)
2
( p, j)  (5, 2)

-classics
(10,1)
t  60
color-quantum
black-

experiment
classical
Farey Rule
0
1
0
1
1
2
0
1
0
1
1
1
1
1
1
1
1
4
1
3
1
2
2
3
1
3
1
2
2
3
j
 ( p, j )
p
3
4

1
1
Boundary of existence of periodic orbits
j
k  2 p  
p
Boundary of stability
width of tongue
1
3/ 2
p
km
“size” of tongue decreases with
p
Farey hierarchy natural
1
p
After 30 kicks
k
  0.3902..
k
Tunneling out of Phase Space
Islands of Maps
Resonance Assisted Decay of
Phase Space Islands
Numerical data
Analytical
approximation
Continuum
formula
n0  0
(ground state)
Effects of Interatomic Interactions
1 2
H =  pˆ   x  k cos x   (t  m)
2
m
 TG 2 
 

M


k

2
 MT 
g
G 
 

i  ( x, t )   H  u ( x, t ) 2  ( x, t )
t
t  25
linear
focusing
#
defocusing

t  45
t  45
repulsive
linear
attractive
t  45
maximum
position
momentum
initial
t  45
initial
t  45
u  3
linear
initial
focusing
#
defocusing

Stability
Probability inside island
|u 3
Number of non-condensed particles
Summary of results
1.
Fictitious classical mechanics to describe
quantum resonances takes into account
quantum symmetries: conservation of
quasimomentum and
i n2l
i nl
e
e
2. Accelerator mode spectroscopy and the Farey
hierarchy
3. Islands stabilized by interactions
4. Steps in resonance assisted tunneling