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