1. Quantum dynamics with ultra cold atoms Nir Davidson Weizmann Institute of Science Billiards BEC I. Grunzweig, Y. Hertzberg, A. Ridinger (M. Andersen, A. Kaplan) Eitan Rowen, Tuesday

2. R E 1 nm Dynamics inside a molecule: quantum dynamics on nm scale Fsec laser pulse

3. Is there quantum chaos? Classical chaos: distances between close points grow exponentially Quantum chaos: distance between close states remains constant Asher Peres (1984): distance between same state evolved by close Hamiltonians grows faster for (underlying) classical chaotic dynamics ??? 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)

4. 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

5. Wedge billiards: chaotic and mixed phase space

6. Criteria for “quantum” to “classical” transition Old: large state number Quantum dynamics with ~10 6 : 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 New: “mixing” to many states by small perturbation But “no mixing” is hard to get

7. Pulsed microwave spectroscopy Prepare Atomic Sample → MW-pulse Sequence → Detect Populations Off On cooling and trapping ~10 6 rubidium atoms optical pumping to π-pulse: π/2-pulse: optical transition MW “clock” transition

8. Ramsey spectroscopy of free atoms H = H int + H ext → Spectroscopy of two-level Atoms π/2 T MW Power Time

9. Ramsey spectroscopy of trapped atoms  E HF |1,Ψ> |2,Ψ> |1,Ψ> H2H2 H1H1 e -iH 2 t |2,Ψ> e -iH 1 t |1,Ψ> … Microwave pulse General case: Nightmare Short strong pulses: OK (Projection)

10. Ramsey spectroscopy of single eigenstate π/2 MW Power Time T For small Perturbation:

11. Ramsey spectroscopy of thermal ensemble π/2 MW Power Time T Averaging over the thermal ensemble destroys the Ramsey fringes For small Perturbation:

12. Echo spectroscopy (Han 1950) π/2 T MW Power Time π T t=T t=2T NOTE: classically echo should not always work for dynamical system !!!!

13. Echo spectroscopy π/2 T MW Power Time π T Coherence De-Coherence Ramsey Echo BUT: it works here !!!!

14. Ramsey Echo Echo vs. Ramsey spectroscopy H2H2 H 1 H2H2 H1H1 H 2

15. Quantum dynamics in Gaussian trap Coherence De-Coherence Calculation for H.O. T osc /2T osc  E HF

16. Long-time echo signal Coherence De-Coherence 2-D: 1-D:

17. Observation of “sidebands” Π-pulse 4π-pulse

18. Quantum stability in atom-optic billiards ~10 4

19. Quantum stability in atom-optic billiards ~10 4 D. Cohen, A. Barnett and E. J. Heller, PRE 63, 046207 (2001)

20. Avoid Avoided Crossings

21. Quantum dynamics in mixed and chaotic phase-space Coherent  Incoherent  Perturbation strength Perturbation-independent decay

22. Quantum dynamics in perturbation-independent regime

23. Shape of perturbation is also important

24. … and even it’s position

25. No perturbation-independence

26. Finally: back to Ramsey (=Loschmidt)

27. 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)

28. Control classical dynamics (regular, chaotic, mixed…) Quantum dynamics with ~10 6 ???? TzahiArielNir Atom Optics Billiards

29. 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