Entanglement and Coherence in Spin-1 Bose Gas --

1. Entanglement and Coherence in Spin-1 Bose Gas -- an exact solution for quantum dynamics of a many-body system Jason Ho The Ohio State University International Conference in Recent Progress in Quantum Mechanics and Its Applications Chinese University of Hong Kong December 14, 2005

2. Recent Progress in Quantum Mechanics and Its Applications (1) Phase coherence, entanglement, (2) Novel condensed matter systems/phenomena Use BEC to illustrate these progress and application

3. Hallmark of BEC : phase coherence Off-diagonal long range order -- coherence at different location

4. Penrose-Onsager characterization of Bose condensation Existence of a single macroscopic eigenvalue in : Eigenvalues : Eigenfunctions

5. Off-Diagonal Oder Phase Coherence CN Yang, RMP, ORLRO paper + …

6. Other possibilities ? Nozieres, 1992 Fragmented condensate Strongly correlated

7. Possible type of ground states: coherent Fragmented Strongly correlated

8. Single condensate state of spin-F Bosons Single particle density matrix contains a single eigenvalue of order N Spin-1/2 Spin-1 scalar

9. How to change phase coherence in a controlled way?

10. Simple Example : Pseudo-spin 1/2 system 1 2 t Hilbert space:

11. BEC --> Fragmentation Coherence --> squeezing --> entanglement 1 2 t

12. BEC --> Fragmentation Coherence --> squeezing --> entanglement 1 2 t

13. BEC --> Fragmentation Coherence --> squeezing --> entanglement 1 2 t ,

14. BEC --> Fragmentation Coherence --> squeezing --> entanglement 1 2 t ,

15. BEC --> Fragmentation Coherence --> squeezing --> entanglement 1 2 t ,

16. Schrodinger Cat State: huge number fluctuation => very delicate Coherent State : very robust If there is a robust mechanism that can Convert the coherent state into a Schrodinger Cat state from time to time, then can realize these cat state at these specific time. Answer : Use Bosonic enhancement

17. Spin-1 Bose gas Consider a term like this :

18. Spin-1 Bose gas Consider a term like this :

19. Spin-1 Bose gas Consider a term like this :

20. Spin-1 Bose gas Consider a term like this :

21. Spin-1 Bose gas Consider a term like this :

22. Dynamical Evolution of Spin-1 Bose Gas

23. N spin-1 Boson in a single level Effect of spin degeneracy on BEC Spin-1 Bose Gas A deep harmonic trap

24. Spin dynamics of spin-1 Bose gas Spin-1 Bose Gas A deep harmonic trap Hilbert space

26. Effect of spin degeneracy on BEC Effect of spin degeneracy on BEC Spin-1 Bose Gas A deep harmonic trap Under spin rotation, rotates like a 3D Cartesean vector . : 3D rotation

27. C>0 = Exact ground state :

28. Spin dynamics of spin-1 Bose gas 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Spin-1 Bose Gas 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 A deep harmonic trap Chapman et.al. PRL 04, PRL 05 Watch change with time

29. Data in Chapman et.al. PRL 04 Oscillatory behavior in Large fluctuations Small fluctuation

30. 0.1% magnetization Large fluctuation Oscillatory

32. Hilbert space :

34. is a sum of two theta functions !

35. Elliptic Functions are mathematicians' fairy land. Those who once glazed upon them are forever captured.

37. Relate to those at

38. Relate to those at

40. time

43. N=1000

44. Generality of the quantum Carpet

45. Periodic generation of Cat state and coherent state. • Self similar in “space-time”. • Quantum dynamics (periodic revival) can be • revealed at short times. • **Fluctuation in Chapman’s experiment is physical. • **Specific predictions to reveal quantum dynamics Printing and deprinting phases on the quantum carpet

46. New Era in ultra-cold atoms Strongly correlated cold atoms Engineering of quantum states

47. N=100 N=1000

49. Effect of spin degeneracy on BEC Effect of spin degeneracy on BEC Spin-1 Bose Gas A deep harmonic trap : spin-1 matrix

50. Effect of spin degeneracy on BEC Spin-1 Bose Gas A deep harmonic trap : Antiferromagnetic : Ferromagnetic , What is the ground state for c>0 ?