quantum chaos and atom optics : from experiments to number theory
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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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.
,Advice and comments: M.V. Berry, Y. Gefen, M. Raizen, W. Phillips, D. Ullmo, P.Schlagheck, E. Narimanov
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
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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
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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
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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
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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
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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 0( , )p Accelerated , also vicinity accelerated
Robust , holds also for vicinity of 2K p
kick/ 2
2p
t
0
1
2
/ 2
/ 2 2
/ 2 4
/ 2 2 m
0
1
2
0
2
4
2m
p
p
p
p m
2
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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
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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
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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
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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
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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
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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
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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
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Effect of Gravity on Kicked Atoms
Quantum accelerator modes
A short wavelength perturbation superimposed on long wavelength behavior
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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
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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
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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
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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
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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”
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-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
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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
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Color --- Husimi (coarse grained Wigner) -classicsblack
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Color-quantum Lines classical
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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
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Color-quantum Lines classical
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decay rate
transient
decay mode
tP e
/Ae
/| |Ae
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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
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(10,1)( , ) (5, 2)p j -classics
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color-quantum
black- classical
60t
experiment
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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
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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
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After 30 kicks
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k
0.3902..
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k
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Tunneling out of Phase Space Islands of Maps
Resonance Assisted Decay of Phase Space Islands
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Numerical data
Analytical approximation
0 0n (ground state)
Continuum formula
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Effects of Interatomic Interactions
21ˆ cos ( )
2 m
p x k x t m =H
2TG
M
2
k
MTg
G
2( , ) ( , ) ( , )i x t x t x tt
u H
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linear
#
focusing
defocusing
25t
45t
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linear attractiverepulsive45t
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position momentum
45t maximum
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45t initial
initial
initial
linear
#
focusing
defocusing
45
3
t
u
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Probability inside island
| 3u Number of non-condensed particles
Stability
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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
hierarchy3. Islands stabilized by interactions4. Steps in resonance assisted tunneling
2i n l i nle e