quantum chaos and atom optics : from experiments to number theory

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

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 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

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

Tunneling out of Phase Space Islands of Maps

Resonance Assisted Decay of Phase Space Islands

Numerical data

Analytical approximation

0 0n (ground state)

Continuum formula

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

linear

#

focusing

defocusing

25t

45t

linear attractiverepulsive45t

position momentum

45t maximum

45t initial

initial

initial

linear

#

focusing

defocusing

45

3

t

u

Probability inside island

| 3u Number of non-condensed particles

Stability

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