1. Quantum Chaos and Atom Optics: from Experiments to Number Theory Italo Guarneri, Laura Rebuzzini, Sandro Wimberger and S.F. Advice and comments: M.V. Berry, Y. Gefen, M. Raizen, W. Phillips 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

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) Effects of gravity, Oxford 1999 New resonance Fictitious Classical mechanics Far from the classical limit (2002) Quantum nonlinear resonance Short wavelength perturbation

4. 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 2. Driving Electric fielddipole potential On center of mass 3. Detection of momentum distribution

5. relative to free fall any structure? =momentum 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?? Experimental results

6. Kicked Rotor Model Dimensionless units

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

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

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

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

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

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

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

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

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

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

17. NOT periodicquasimomentum NOT conserved gauge transformation to restore periodicity integer introduce fictitious classical limit whereplays the role of

18. Gauge Transformation same classical equation for Formomentum relative to free fall quasimomentumconserved

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

20. “momentum” quantization effective Planck’s constant dequantization Fictitious classical mechanics useful fornear resonance destroys localization dynamics of a kicked system whereplays the role of meaningful “classical limit”

21. -classical dynamics motion on torus change variables

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

24. Color --- Husimi (coarse grained Wigner) -classicsblack

25. Color-quantum Lines classical

26. relative to free fall any structure? =momentum 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?? Experimental results

27. Color-quantum Lines classical

28. decay rate transient decay mode

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

30. -classics

31. color-quantum black- classical experiment

33. Farey Rule

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

35. After 30 kicks

38. 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

39. General Context Accelerator mode (b) Measurement of 1.How general are the robust resonances? 2.Experimental preparations of coherent superpositions 3.Manipulation of resonances and interferometry (a) Narrow coherent momentum distribution 4. Tuning “gravity” 5. Resonaces and number theory? 6. Improved resolution of ?? 7. Quantum ratchets??

40. Resonances NO gravity

41. Momentum distribution at resonance at resonance using up to constants Exactly solvable, typically localized states. resonances and ballistic motion for specific quasimomentum for example Effectively ballistic motion for a timefor an interval of size in

42. M.F. Andersen, A. Varizi, M.B. d’Arcy, J.M. Grossman, K. Helmerson, W.D. Phillips + simulationtheory experiment NIST2005

43. resonance

44. Dynamics near resonance Quantum resonanceClassical resonance What is? at resonance But averaged over a wide range of quasimomentum mod Average overand over scaling,dependence only via

45. M.F. Andersen, A. Varizi, M.B. d’Arcy, J.M. Grossman, K. Helmerson, W.D. Phillips 1 NIST2005 scaling,dependence only via

46. Averaging over Qusimomentum

47. -classical quantum

48. Experiment on Cesium Atoms (Wimberger, Sadgrove, Parkins, Leonhardt, PRA 71, 053404 (2005))

49. Experiment on Cesium Atoms (Wimberger,Sadgrove, Parkins, Leonhardt, PRA 71, 053404 (2005))

50. Summary of results 1.Fictitious classical mechanics to describe quantum resonances takes into account quantum symmetries: conservation of quasimomentum and 2. -classical description of quantum resonances and their vicinity, relation to classical resonances 3. Scaling theory for vicinity of resonance (averaged and not averaged over Quasimomentum) 4. Narrow as peaks near resonance ( Found to hold also for higher order resonances) 5. Momentum distribution functions at resonance 6. Comparison with experiments (general characteristics also for higher order) ????Theory for Higher Order Resonances ????? Dana and coworkers