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Experiment
F.L. Moore, J.C. Robinson, C.F. Bharucha, B. Sundaram and M.G. Raizen, Phys. Rev. Lett. 25, 4598 (1995)
1. Laser cooling of Na Atoms
2. Driving
e
g
L
d E
E
Electric field
potential
dipole
E  d  E 2
On center of mass
x
3. Detection of momentum distribution
V  cos Gx  (t  mT )
m
R. Blumel, S. Fishman and U. Smilansky, J. Chem. Phys., 84, 2604 (1986).
Figure 2: Average energy of the rotor
as a function of time for k=2 and
Δ(t)=Δ(N=7) (t). (a) Quantum mechanical
calculations for the localized (t =2)
and extended (t =2π/3) case, (b)
Classical calculation (t =2).
R. Blumel, S. Fishman and U. Smilansky, J.
Chem. Phys., 84, 2604 (1986).
Figure 3
R. Blumel, S. Fishman and U. Smilansky, J. Chem. Phys., 84, 2604 (1986).
Figure 4: Some quasienergy states
characterized by a large
overlap with the rotor
ground state |0> for
interaction strength k=2
(a) t =2, (b) t =2π/3.
R. Blumel, S. Fishman and U. Smilansky,
J. Chem. Phys., 84, 2604 (1986).
Experiment
F.L. Moore, J.C. Robinson, C.F. Bharucha, B. Sundaram and M.G. Raizen, Phys. Rev. Lett. 25, 4598 (1995)
1. Laser cooling of Na Atoms
2. Driving
e
g
L
d E
E
Electric field
potential
dipole
E  d  E 2
On center of mass
x
3. Detection of momentum distribution
V  cos Gx  (t  mT )
m
pˆ 2
H =
 K cos x  (t  mT )
2M
m
1 2
H = t pˆ  k cos x  (t  m)
2
m
kicked rotor
0   = x  2
classical
K = kt
typical K
K  2 l
diffusion in
t = typical
t / 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)
Moore, … Raizen
PRL 75, 4598 (1995)
Observed localization
Indeed Quantum
For values of K where there are accelerator
Modes – No exponential localization
Remember motion bounded in momentum
Klappauf…. Raizen PRL 81, 4044 (1998)
Effect of Gravity on Kicked Atoms
Quantum accelerator modes
A short wavelength perturbation superimposed on long wavelength behavior
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?
t / 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??
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 = t pˆ   x  k cos x  (t  m)
2
m
 TG 2 
t =

M


k=
in experiment
x
NOT periodic

2
k 
 MT 
 =
g
G 
  0.1
quasimomentum NOT conserved
x
NOT periodic
quasimomentum NOT conserved
gauge transformation to restore periodicity
t = 2 l  
l=

integer
introduce fictitious classical limit where

1
plays the role of
Gauge Transformation
1 2
H I = t pˆ   x  k cos x  (t  m)
2
m
1
2
H II = t ( pˆ  t   k cos x  (t  m)
2
m
same classical equation
for
x

i  = H I
t
 i xt
 ( x, t ) = e  ( x, t )

i  = H II 
t
For H II momentum relative to free fall (  t )
quasimomentum 
conserved
p = n

n̂ = i
 = x mod(2 )

Uˆ = Uˆ kick  Uˆ free
Quantum Evolution
ik cos
ˆ
U kick = e
Uˆ free = e
Uˆ free = e
Uˆ free = e
1

 i  t nˆ 2  nˆt (  t  / 2  
2

t = 2 l  
up to terms independent of
operators but depending on
 i n2l
e
 1 ˆ2 ˆ
 i   n  n l  nˆt (  t  / 2  
2

i   ˆ2 ˆ

  I  I ( l t (  t  / 2)  
| |  2

Uˆ kick = e
“momentum”
i
k
| |
cos
=e
 i nl
 = 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 (   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t   l  t  t / 2
Jt = It   tt   l  t  t / 2
Jt 1 = Jt  k sin t 1 t
motion on torus
 =  mod(2 )
 t 1 =  t  J t
J = J mod(2 )
Accelerator modes
Jt 1 = Jt  k sin t 1 t
motion on torus
 t 1 =  t  J t
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 = t / k
0
accelerator mode
n = n0  tt / 

Color --- Husimi (coarse grained Wigner)
black
-classics

Color-quantum
Lines
classical
Experimental results
relative to free fall
any
structure?
t / 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 t
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
t
=
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
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