Quantum Chaos and Atom Optics : from Experiments to Number Theory

1. Quantum Chaos andAtom Optics: from Experiments to Number Theory Italo Guarneri, Laura Rebuzzini, Michael Sheinman Sandro Wimberger, Roberto Artuso and S.F. 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 , Advice and comments: M.V. Berry, Y. Gefen, M. Raizen, W. Phillips, D. Ullmo, P.Schlagheck, E. Narimanov

2. Quantum Chaos Atom Optics Kicked Rotor Classical Diffusion (1979 ) Quantum Deviations from classical behavior Anderson localization (1958,1982) Observation of Anderson localization for laser cooled Cs atoms (Raizen, 1995) Fictitious Classical mechanics Far from the classical limit (2002) Effects of gravity, Oxford 1999 New resonance Quantum nonlinear resonance Short wavelength perturbation

4. 2. Driving Electric field • dipole potential Experiment R.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 On center of mass 3. Detection of momentum distribution

5. Experimental results relative to free fall any structure? Accelerator mode What is this mode? Why is it stable? What is the decay mechanism and the decay rate? Any other modes of this type? How general?? =momentum

6. Kicked Rotor Model Dimensionless units

7. Assume Robust , holds also for vicinity of Classical Motion Standard Map kick Accelerated , alsovicinity accelerated

8. Effectively random kick Diffusion in kick kick For values of Where acceleration , it dominates Nonlinearity Accelerator modes robust For typical kick

9. For typical Effectively random Diffusion in for integer some for example Classical Motion Standard Map Diffusion Acceleration and vicinity accelerated

10. Andersonlocalization Evolution operator rational irrational Quantum resonance Anderson localization like for 1D solids with disorder pseudorandom Quantum

11. pseudorandom irrational Simple resonances: rational Quantum resonance classical Talbot time Eigenstates of Exponentially localized quantum Quantum Anderson localization like for 1D solids with disorder

12. rotor Quantum : Not quantized rational, resonance only for few values of periodic transitions (quasimomentum) CONSERVED fractional part of Kicked Particle Classical-similar to rotor irrational Anderson localization classical quantum

13. kicked rotor kicked particle typical diffusion in classical diffusion in acceleration acceleration integer arbitrary quantum typical Localization in Localization in resonances resonances only for fewinitial conditions rational

14. 2 (momentum) < momentum F.L. Moore, J.C. Robinson, C.F. Bharucha, B. Sundaram and M.G. Raizen, PRL 75, 4598 (1995)

15. Effect of Gravity on Kicked Atoms Quantum accelerator modes A short wavelength perturbation superimposed on long wavelength behavior

16. dimensionless units in experiment NOT periodic quasimomentum NOT conserved Experiment-kicked atoms in presence of gravity

17. integer introduce fictitious classical limit where plays the role of NOT periodic quasimomentum NOT conserved gauge transformation to restore periodicity

18. same classical equation for For momentum relative to free fall quasimomentum conserved Gauge Transformation

19. up to terms independent of operators but depending on Quantum Evolution “momentum”

20. “momentum” dynamics of a kicked system where plays the role of effective Planck’s constant Fictitious classical mechanics useful for near resonance dequantization quantization destroyslocalization meaningful “classical limit”

21. motion on torus -classical dynamics change variables

22. motion on torus -classical periodic orbit stable period 1 (fixed points): quantum accelerator mode solution requires choice of and accelerator mode Accelerator modes Solve for stable classical periodic orbits follow wave packets in islands of stability

24. Color --- Husimi (coarse grained Wigner) black -classics

25. Color-quantum Lines classical

26. Experimental results relative to free fall any structure? Accelerator mode What is this mode? Why is it stable? What is the decay mechanism and the decay rate? Any other modes of this type? How general?? =momentum

27. Color-quantum Lines classical

28. decay mode decay rate transient

29. period Acceleration proportional to differencefrom rational fixed point (period, jump in momentum) Higher accelerator modes: observed in experiments as Farey approximants of gravity in some units Accelerator mode spectroscopy map: motion on torus

30. -classics

31. color-quantum black- classical experiment

33. Farey Rule

34. “size” of tongue decreases with width of tongue Boundary of existence of periodic orbits Boundary of stability Farey hierarchy natural

35. After 30 kicks

38. Tunneling out of Phase Space Islands of Maps Resonance Assisted Decay of Phase Space Islands

40. Numerical data Analytical approximation Continuum formula (ground state)

41. Effects of Interatomic Interactions

42. linear focusing defocusing

43. linear repulsive attractive

44. maximum position momentum

45. initial linear focusing defocusing initial initial

46. Stability Probability inside island Number of non-condensed particles

47. 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 3. Islands stabilized by interactions 4. Steps in resonance assisted tunneling