Quantum Chaos and Atom Optics: from Experiments to Number Theory Italo Guarneri, Laura Rebuzzini,...

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

-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..

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

Quantum Chaos and Atom Optics: from Experiments to Number Theory Italo Guarneri, Laura Rebuzzini,...

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