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ExperimentF.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 E
d E
Electric field dipole
potential 2E d E
xcos ( )
m
V Gx t mT
On center of mass
3. Detection of momentum distribution
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 2) and extended 2π/3) case, (b) Classical calculation 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 quasi-energy states characterized by a large overlap with the rotor ground state |0> for interaction strength k=2 (a) =2, (b) =2π/3.
R. Blumel, S. Fishman and U. Smilansky, J. Chem. Phys., 84, 2604 (1986).
ExperimentF.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 E
d E
Electric field dipole
potential 2E d E
xcos ( )
m
V Gx t mT
On center of mass
3. Detection of momentum distribution
2ˆcos ( )
2 m
pK x t mT
M H
21ˆ cos ( )
2 m
p k x t m H =
kicked rotor0 2x
kicked particlex
typical K diffusion in p diffusion in p
2K l accelerationacceleration
p integer p arbitrary
p p n typical
Localization in pLocalization in p
/ 2 rational resonances resonances only for few initial conditions
classical
K k
quantum
F.L. Moore, J.C. Robinson, C.F. Bharucha, B. Sundaram and M.G. Raizen, PRL 75, 4598 (1995)
tmomentum
2
2
2
kt
(momentum)1
22
<
t
Moore, … Raizen PRL 75, 4598 (1995)
Observed localizationIndeed Quantum
For values of where there are acceleratorModes – No exponential localization
K
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
ExperimentR.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 E
d E
Electric field dipole
potential 2E d E
x
Mgx
cos ( )m
V Gx t mT On center of mass
3. Detection of momentum distribution
relative to free fall
any structure?
/ 2 1 67 s
p=momentum
Accelerator modeWhat is this mode?Why is it stable?What is the decay mechanism and the decay rate?Any other modes of this type?How general??
Experimental results
Experiment-kicked atoms in presence of gravity
2
1 cos ( )2 2 m
pGx t mT
MMgx
H
4 /G 895nm 66.5T s l
dimensionless units Gx x /t T t H
in experiment k 0.1
21ˆ cos ( )
2 m
p k x mx t H =
2TG
M
2k
MT
gG
x NOT periodic quasimomentum NOT conserved
x NOT periodic quasimomentum NOT conserved
gauge transformation to restore periodicity
2 l l integer 1
introduce fictitious classical limit where plays the role of
Gauge Transformation
21ˆ cos ( )
2 m
p k x mx t IH =
21ˆ cos ( )
2 m
p t k x t m IIH =
same classical equation for x
it
it
I
II
H
H( , ) ( , )i xtx t e x t
For IIH momentum relative to free fall ( )t
mod(2 )
p
x
n
quasimomentum conserved
n̂ i
Quantum Evolution ˆ ˆ ˆkick freeU U U
cosˆ ikkickU e
21ˆ / 2
2ˆ
ˆi n t
ree
n
fU e
2 l 2i n l i nle e
21ˆ / 2
2ˆ ˆ
ˆni n t
fre
l
e
n
U e
ˆ ˆ| | | |I n i
“momentum”
( )sign 2ˆ ˆ ( / 2)
| | 2ˆI I
il t
freeU e
|cos|ˆ
i
kick
k
U e
| |k k
up to terms independent ofoperators but depending on
ˆ | |I i
“momentum” | |k k
quantization p ix
21/ 2ˆ
2ˆ ( )lI I t H
cos| | | |ˆk ii
U e e
H
| | effective Planck’s constant
dequantization | |i I
Fictitious classical mechanics useful for | | 1 near resonance
destroys localization
dynamics of a kicked system where | | plays the role of
meaningful “classical limit”
-classical dynamics
1 1sint t tI I k 1 / 2t tt lI t
/ 2t tJ I lt
1 1sint t tJ J k 1t t tJ
motion on torus mod(2 ) mod(2 )J J =
cos ( )m
k t m H =H
change variables
Accelerator modes
1 1sint t tJ J k 1t t tJ
motion on torus mod(2 ) mod(2 )J J =Solve for stable classical periodic orbits follow wave packets in islands of stability
quantum accelerator mode stable -classical periodic orbit
period 1 (fixed points): 00J 0sin / k
solution requires choice of and 0
accelerator mode 0 /n n t
Color --- Husimi (coarse grained Wigner) -classicsblack
Color-quantum Lines classical
relative to free fall
any structure?
/ 2 1 67 s
p=momentum
Accelerator modeWhat is this mode?Why is it stable?What is the decay mechanism and the decay rate?Any other modes of this type?How general??
Experimental results
Color-quantum Lines classical
decay rate
transient
decay mode
tP e
/Ae
/| |Ae
Accelerator mode spectroscopy
period pfixed point
0
0
2
2
p
p
J J j
n
/ | |n I
0
2 | |
| |
jn n t
p
Higher accelerator modes: ( , )p j (period, jump in momentum)
observed in experiments
motion on torus
1 1sint t tJ J k 1t t tJ map:
/j p as Farey approximants of mod(1)2
gravity in some units
Accelerationproportional to
difference from rational
(10,1)( , ) (5, 2)p j -classics
color-quantum
black- classical
60t
experiment
Farey Rule1
1
1
3
2
3
1
4
3
4
0
10
1
0
1
0
1
1
11
1
1
1
1
2
1
2
1
21
3
2
3
( , )
jp j
p
Boundary of existence of periodic orbits
2j
k pp
Boundary of stability
width of tongue1
p
3/ 2
1mk p
“size” of tongue decreases with p
Farey hierarchy natural
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
2. Accelerator mode spectroscopy and the Farey
hierarchy
2i n l i nle e