Quantum Chaos and Atom Optics: from Experiments to Number Theory
Italo Guarneri, Laura Rebuzzini,
Sandro Wimberger and S.F.
Advice and comments: M.V. Berry, Y. Gefen, M. Raizen, W. Phillips
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
Quantum Chaos Atom Optics
Kicked RotorClassical Diffusion (1979 )Quantum Deviations from classical behavior Anderson localization (1958,1982)
Observation of Anderson localization for laser cooled Cs atoms (Raizen, 1995)
Effects of gravity, Oxford 1999New resonance
Fictitious Classical mechanics Far from the classical limit (2002)
Quantum nonlinear resonance
Short wavelength perturbation
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
Kicked Rotor Model
F
F
n i
21ˆ cos ( )
2 m
n k t m H =
22ˆ cos ( )
2 m
n K t mTI
H
T
I Dimensionless units
Kk
Classical Motion
m 1m
tmp 1mp m 1m
( p n )K k
1
1 1sinm m m
m m m
p
p p K
Standard MapAssume 2K
0
1
2
/ 2
/ 2 2
/ 2 4
/ 2 2 m
0
1
2
0
2
4
2m
p
p
p
p m
0 0( , )p Accelerated , also vicinity accelerated
Robust , holds also for vicinity of 2K p
kick/ 2
2p
t
For typical 1K
kick
kick
kick
kick
sin mEffectively random
Diffusion in p2p
t
For values of K Where acceleration , it dominates
Nonlinearity Accelerator modes robust
0t 0t
p
Classical Motion ( p n )K k
1
1 1sint t t
t t t
p
p p K
Standard Map
2p
t
2p
t
For typical 1K sin t Effectively random Diffusion in p
for 2K integer
Diffusion
Acceleration
( / 2,0)0 0( , )p for example some
and vicinity accelerated
0t 0t
Quantum T
I
21ˆ cos ( )
2 m
n k t m H =
2ˆcos2ˆ i n ikU e e
1
ˆt tU Evolution operator
2
rational Quantum resonance 2 2p t
2
irrational
2ˆ2i n
e
pseudorandom Anderson localization like
for 1D solids with disorder
Anderson localization
Quantum T
I
21ˆ cos ( )
2 m
n k t m H =
classical
quantum
Eigenstates of
Exponentially localizedU
2n
t
Anderson localization like for 1D solids with disorder
/ 2 rational Quantum resonance 2p
tSimple resonances: 2 ,4 ...2 l
4 Talbot time
/ 2 irrational2ˆ
2i n
e
pseudorandom
Kicked Particle
rotor21ˆ cos ( )
2 m
p k x t m H =
Classical-similar to rotorQuantum : x p Not quantized
cos x periodic transitions p p n fractional part of p (quasimomentum ) CONSERVED
/ 2 rational, resonance only for few values of
classical
quantum
2p
t
p
( )P p tAnderson localization / 2 irrational
( )V x
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
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
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
2
k
MTg
G
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
n 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
General Context
Accelerator mode
Accelerator mode
Accelerator mode
(b) Measurement of g
1. How general are the robust resonances?
2. Experimental preparations of coherent superpositions
3. Manipulation of resonances and interferometry
(a) Narrow coherent momentum distribution
4. Tuning “gravity”5. Resonaces and number theory?6. Improved resolution of ??7. Quantum ratchets??
Resonances NO gravity
Momentum distribution at resonance 2 l 2ˆ
cos 2ˆ i nikU e e
at resonance using 2i n l i nle e
up to constants
ˆ2 1cosˆ i l nikU e e Exactly solvable, typically localized states. resonances and ballistic motion for specific quasimomentum for example 1/ 2
Effectively ballistic motion for a time t for an interval of size 1/ t in
2p t
2l M.F. Andersen, A. Varizi, M.B. d’Arcy, J.M. Grossman, K. Helmerson, W.D. Phillips
N t +simulation theory experiment
NIST2005
resonance
Dynamics near resonance 2 l
Quantum resonance Classical resonanceWhat is ,tE ?
21
4tE k t
at resonance
But averaged over a wide range of quasimomentum
( 0)
0J l mod 2Average over 0 and over
scaling, t dependence only via | |res
tx t k
t
The first order resonance
0
0.2
0.4
0.6
0.8
1
1.2
0 2 4 6 8 10
x
R(x
) N=16
N=8
Theory
2l M.F. Andersen, A. Varizi, M.B. d’Arcy, J.M. Grossman, K. Helmerson, W.D. Phillips
N t1
NIST2005
,t
t
ER
E
scaling, t dependence only via | |res
tx t k
t
Averaging over Qusimomentum
( )H x -classical quantum
0
,t
t
ER
E
Experiment on Cesium Atoms (Wimberger, Sadgrove, Parkins, Leonhardt, PRA 71, 053404 (2005))
Experiment on Cesium Atoms (Wimberger,Sadgrove, Parkins, Leonhardt, PRA 71, 053404 (2005))
Summary of results1. Fictitious classical mechanics to describe quantum resonances takes into account quantum symmetries: conservation of quasimomentum and
2. -classical description of quantum resonances and their vicinity, relation to classical resonances
3. Scaling theory for vicinity of resonance (averaged and not averaged over Quasimomentum)
4. Narrow as peaks near resonance ( Found to hold also for higher order resonances)
5. Momentum distribution functions at resonance
6. Comparison with experiments (general characteristics also for higher order)
2i n l i nle e
21/ t
????Theory for Higher Order Resonances ?????Dana and coworkers
