1. Spectral relationships between kicked Harper and on-resonance double kicked rotor operators Collaborators:Jiao Wang and Jiangbin Gong Speakers:Anders S. Mouritzen (that’s me) Wayne Lawton (coming soon) arXiv: 0807.4276v1 [math-ph]

2. What this is about: We have:1) Kicked Harper Model An important theoretical model. No existing experimental realizations. 2) Kicked Rotor Model Another important theoretical model. Experimentally realizable. Spectral relationships between kicked Harper and on-resonance double kicked rotor operators

3. What this is about: We have:1) Kicked Harper Model An important theoretical model. No existing experimental realizations. 2) Kicked Rotor Model Another important theoretical model. Experimentally realizable. We wish to: Experimentally realize 1) using 2). Spectral relationships between kicked Harper and on-resonance double kicked rotor operators

4. What this is about: We have:1) Kicked Harper Model An important theoretical model. No existing experimental realizations. 2) Kicked Rotor Model Another important theoretical model. Experimentally realizable. We wish to: Experimentally realize 1) using 2). What we get: An experimentally realizable operator with the same spectrum as 1), but with different dynamics. Spectral relationships between kicked Harper and on-resonance double kicked rotor operators

5. A closer look at the models: Kicked Harper operator: On-resonance double kicked rotor operator: Spectral relationships between kicked Harper and on-resonance double kicked rotor operators Use periodic boundary conditions. Basis:

6. Suggested experimental realization: Spectral relationships between kicked Harper and on resonance double kicked rotor operators Cold atomic gas or Bose-Einstein condensate

7. Suggested experimental realization: Spectral relationships between kicked Harper and on resonance double kicked rotor operators Laser Cold atomic gas or Bose-Einstein condensate

8. Suggested experimental realization: Spectral relationships between kicked Harper and on resonance double kicked rotor operators Laser A pulsed, standing wave sets up a potential energy:

9. Suggested experimental realization: Spectral relationships between kicked Harper and on resonance double kicked rotor operators Laser A pulsed, standing wave sets up a potential energy:

10. Suggested experimental realization: Spectral relationships between kicked Harper and on resonance double kicked rotor operators Laser The operator U ordkr is periodic in x with period 1. Assumption: The state also has this periodicity. We only have to look at one such period in

11. Suggested experimental realization: Spectral relationships between kicked Harper and on resonance double kicked rotor operators tjtj t j + α T R / 2 Laser intensity Time (1-α / 2)T R TRTR t j+1 = t j + T R

12. Suggested experimental realization: Spectral relationships between kicked Harper and on resonance double kicked rotor operators Free propagator: Laser intensity Time tjtj t j + α T R / 2t j+1 = t j + T R Note, that:

13. Suggested experimental realization: Spectral relationships between kicked Harper and on resonance double kicked rotor operators Laser potential: Neglect kinetic energy. Laser intensity Time tjtj t j + α T R / 2t j+1 = t j + T R

14. Suggested experimental realization: Spectral relationships between kicked Harper and on resonance double kicked rotor operators Estimates of obtainable experimental parameters:

15. Let’s look at the spectrums, σ: Spectral relationships between kicked Harper and on resonance double kicked rotor operators

16. Why are the spectrums different? Spectral relationships between kicked Harper and on resonance double kicked rotor operators The spectrum is: 1) Discontinuous at rational α for fixed θ (as in the figure below) 2) Continuous and independent of θ at irrational α.

17. Why are the spectrums different? Spectral relationships between kicked Harper and on resonance double kicked rotor operators The spectrum is: 1) Discontinuous at rational α for fixed θ (as in the figure below) 2) Continuous and independent of θ at irrational α. A way out: (due to Hofstadter) Something’s wrong with the physics! Take the union of the spectrums over θ – this is continuous in α.

18. Let’s look at the spectral unions: Spectral relationships between kicked Harper and on resonance double kicked rotor operators

19. Thanks for now ☺ Spectral relationships between kicked Harper and on resonance double kicked rotor operators Please welcome Wayne.