Quantum dynamics with ultra cold atoms

Nir Davidson

Weizmann Institute of Science

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Billiards BEC86 1010 n 1 n

I. Grunzweig, Y. Hertzberg, A. Ridinger (M. Andersen, A. Kaplan)

Eitan Rowen, Tuesday

R

E 1 nm

Dynamics inside a molecule:quantum dynamics on nm scale

Fsec laser pulse

Is there quantum chaos?

• Classical chaos: distances between close points grow exponentially

• Quantum chaos: distance between close states remains constant

1212 |expexp nnnH

iH

in

Asher Peres (1984): distance between same state evolved by close Hamiltonians grows faster for (underlying) classical chaotic dynamics ???

nnnH

iH

in |expexp 12

Answer: yes….but also depends on many other things !!!

One thing with many names: survival probability = fidelity = Loschmidt echo

R. Jalabert and H. Pastawski, PRL 86, 2490 (2001)

PRL 86, 1518 (2001), PRL 87, 274101(2001), PRL 90 023001 (2003)

…and effects of soft walls, gravity, curved manifolds, collisions…..

Atom-optics billiards:decay of classical time-correlations

Wedge billiards: chaotic and mixed phase space

Criteria for “quantum” to “classical” transition

Old: large state number

'' nnnn 610/ nEE

Quantum dynamics with <n>~106: challenges and solutions:

• Very weak (and controlled) perturbation –optical traps + very strong selection rules

• No perturbation from environment - ultra cold atoms

• Measure mixing – microwave spectroscopy

• Pure state preparation? - echo

11010 66 n

1n

New: “mixing” to many states by small perturbation 1' nnn

But “no mixing” is hard to get

Pulsed microwave spectroscopy

Prepare Atomic Sample → MW-pulse Sequence → Detect Populations

Off

On

2

1

3

• cooling and trapping ~106 rubidium atoms• optical pumping to

π-pulse:

π/2-pulse: 22

12

11

i

21 i

1

optical transition

MW “clock” transition

)0,3,5( 2/1 FmFS

)0,2,5( 2/1 FmFS

Ramsey spectroscopy of free atoms

TΔcos121

P2

2/21 i

2/21 Tiie 1

H = Hint + Hext → Spectroscopy of two-level Atoms

π/2 π/2T

MW

Pow

er

Time

Ramsey spectroscopy of trapped atoms

22H11HH 21 extHH int

EHF

2

1

|1,Ψ>

|2,Ψ>

|1,Ψ>

H2

H1

e-iH2t|2,Ψ>

e-iH1t|1,Ψ>

<Ψ| eiH1

te-iH2

t|Ψ>…

Microwave pulse

Microwave pulse

General case: Nightmare Short strong pulses: OK (Projection)

)/( ALopt IV

Ramsey spectroscopy of single eigenstate

π/2 π/2M

W P

ower

Time

T

For small Perturbation:

Ramsey spectroscopy of thermal ensemble

π/2 π/2M

W P

ower

Time

T

Averaging over the thermal ensemble destroys the Ramsey fringes

For small Perturbation:

Echo spectroscopy (Han 1950)

π/2 π/2TM

W P

ower

Time

π T

t=T

t=2T

NOTE: classically echo should not always work for dynamical system !!!!

Echo spectroscopy

π/2 π/2TM

W P

ower

Time

π T

Coherence

De-Coherence Ramsey

Echo

BUT: it works here !!!!

nH

iH

iH

iH

in

2121 expexpexpexp

nH

iH

in

21 expexp Ramsey

Echo

Echo vs. Ramsey spectroscopy

H2

H1

H2

H1

H1

H2

Quantum dynamics in Gaussian trap

Coherence

De-Coherence

Calculation for H.O.

'n,nδn'n

Tosc/2 Tosc

EHF

2

1

Long-time echo signal

nEEEnn nn /' '

4

2 nn'n121

P

Coherence

De-Coherence

610/ EE•2-D:

•1-D:310/ EE

nEEnn nn /1/' '

Observation of “sidebands”

Π-pulse

4π-pulse

Quantum stability in atom-optic billiards

<n>~104

Quantum stability in atom-optic billiards

<n>~104

D. Cohen, A. Barnett and E. J. Heller, PRE 63, 046207 (2001)

Avoid Avoided Crossings

Quantum dynamics in mixed and chaotic phase-space

Coherent

Incoherent

Perturbation strength

Perturbation-independent decay

Quantum dynamics in perturbation-independent regime

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P2

Time between pulses (s)

Chaotic Mixed

Shape of perturbation is also important

… and even it’s position

No perturbation-independence

Finally: back to Ramsey (=Loschmidt)

•Quantum dynamics of extremely high-lying states in billiards:survival probability = Loschmidt echo = fidelity=dephasing?

• Quantum stability depends on: classical dynamics, type and strength of perturbation, state considered and….

• “Applications”: precision spectroscopy (“clocks”) quantum information

Conclusions

Can many-body quantum dynamics be reversed as well?

(“Magic” echo, Pines 1970’s, “polarization” echo, Ernst 1992)

•Control classical dynamics (regular, chaotic, mixed…)

•Quantum dynamics with <n>~106 ????

Tzahi Ariel Nir

Atom Optics Billiards

Atom Optics Billiards

Positive (repulsive) laser potentials of various shapes.Standing Wave

Trap Beam

• Z direction frozen by a standing wave

• Low density collisions

• “Hole” in the wall probe time-correlation function