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Quantum Chaos and Atom Optics : from Experiments to Number Theory

Quantum Chaos and Atom Optics : from Experiments to Number Theory. Italo Guarneri, Laura Rebuzzini, Sandro Wimberger and S.F. Experiments: M. d’Arcy, G. Summy, M. Oberthaler, R. Godun, Z.Y. Ma.

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Quantum Chaos and Atom Optics : from Experiments to Number Theory

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  1. Quantum Chaos andAtom Optics: from Experiments to Number Theory Italo Guarneri, Laura Rebuzzini, Sandro Wimberger 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

  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

  3. 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

  4. 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

  5. Kicked Rotor Model Dimensionless units

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

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

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

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

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

  11. 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

  12. 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

  13. 2 (momentum) < momentum F.L. Moore, J.C. Robinson, C.F. Bharucha, B. Sundaram and M.G. Raizen, PRL 75, 4598 (1995)

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

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

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

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

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

  19. “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”

  20. motion on torus -classical dynamics change variables

  21. 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

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

  23. Color-quantum Lines classical

  24. 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

  25. Color-quantum Lines classical

  26. decay mode decay rate transient

  27. 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

  28. -classics

  29. color-quantum black- classical experiment

  30. Farey Rule

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

  32. After 30 kicks

  33. 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

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

  35. Resonances NO gravity

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

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

  38. resonance

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

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

  41. Averaging over Qusimomentum

  42. quantum -classical

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

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

  45. Summary of results • Fictitious classical mechanics to describe quantum resonances • takes into account quantum symmetries: conservation of quasimomentum • and • -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

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