1. Atomic Bose-Einstein Condensates Mixtures Introduction to BEC Dynamics: (i) Quantum spinodial decomposition, (ii) Straiton, (iii) Quantum nonlinear dynamics. Self-assembled quantum devices. Statics: (a) Broken symmetry ? (b) Amplification of trap displacement

3. Collaborators: P. Ao Hong Chui Wu-Ming Liu V. Ryzhov Hulain Shi B. Tanatar E. Tereyeva Yu Yue Wei-Mou Zheng

4. Introduction to BEC Optical, and Magnetic traps Evaporative Cooling http://jilawww.colorado.edu/bec/

5. Formation of BEC

6. Slow expansion after 6 msec at T >Tc

7. Mixtures: Different spin states of Rb (JILA) and Na (MIT). Dynamics of phase separation: From an initially homogeneous state to a separated state. Static density distribution

8. Classical phase separation: spinodial decomposition At intermediate times a state with a periodic density modualtion forms. Domains grow and merge at later times.

10. Physics of the spinodial decomposition  2 <0 for small q. From Goldstone’s theorem,  q 2 =0 when q=0. For large enough q,  q 2 >0 q 22 q sd

11. Dynamics: Quantum spinodial state In classical phase separation, for example in AlNiCo, there is a structure with a periodic density modulation called the spinodial decomposition. Now the laws are given by the Josephson relationship. But a periodic density modulation still exists.

12. Densities at different times D. Hall et al., PRL 81, 1539 (1998). Right: |1> Middle:|2> Left: total

13. Intermediate time periodic state: Just like the classical case, the fastest decaying mode from a uniform phase occurs at a finite wavevector. This is confirmed by a linear instability analysis by Ao and Chui.

15. Metastability: Sometimes the state with the periodic density exists for a long time

16. H-J Miesner at al. (PRL 82, 2228 1999)

17. Metastability: Solitons are metastable because they are exact solutions of the NONLINEAR equation of motion Solitons are localized in space. Is there an analog with an EXTENDED spatial structure?---the ``Straiton’’

18. Coupled Gross-Pitaevskii equation U: interaction potential; Gij, interaction parameters

19. A simple exact solution: When all the G’s are the same, a solution exist for, For this case, the composition of the mixture is 1:1.

20. Coupled Gross-Pitaevskii equation U: interaction potential; G, interaction parameters:

21. More Generally, in terms of elliptic functions N 1 /N 2 =(G12-G22)/(G11-G12) for G11>G22>G22 ( correspons to Rb) N 1/ N 2 =1 for G11=G22=G12. This can be related to Na (G11=G12>G22) by perturbation theory.

22. Domains of metastability Exact solutions can be found for the one dimensional two component Gross-Pitaevskii equation that exhibits the periodic density modulation for given interaction parameters only for certain compositions. Exact solutions imply metastability: that the nonlinear interaction will not destroy the state. Not all periodic intermediate states are metastable?

23. Density of component 1: Numerical Results Na, 1D MIT parameters 1:1

24. Total density Na MIT parameters 1:1 Gij are close to each other

25. Phase Separation Instability: Interaction energy: Insight: The energy becomes : Total density normal mode stable. The density difference is unstable when

26. Results from Linear Instability Analysis Period is inversely proportional to the square root of the dimensionless coupling constant. Time is proportional to period squared.

27. Hypothesis of stability: System is stable only for compositions close to 1:1.

28. Quantum nonlinear dynamics: a very rich area Rb 4:1 Periodic state no longer stable Very intricate pattern develops.

30. Self assembled quantum devices For applications such as atomic intereferometer it is important to put equal number of BEC in each potential well.

31. Self-assembled quantum devices Phase separation in a periodic potential. Two length scales: the quantum spinodial wavelength qs and the potential period l=2(a+b).

32. Density distribution of component 1 as a function of time Density is uniform at time t=0. As time goes on, the system evolves into a state so that each component goes into separate wells.

33. How to pick the righ parameters: Linear stability analysis can be performed with the transfer matrix method. In each well we have   j =[A j e ip(x-nl) +B j e - ip(x-nl) ]e i  t Get cos(kl)=cos2qa cos2pb-(p 2 +q 2 )sin2qa sin2pb/2pq.

34. How to pick the right parameters? k=k 1 +ik 2; real wavevector k 1 l (solid line) and imaginary wavevector k 2 l (dashed line) vs  2. Fastest mode occurs when k 1 l ¼ 

35. Topics Quantum phase segregation: domains of metastability and exact solutions for the quantum spinodial phase. The dynamics depends on the final state. What are the final states? Broken symmetry: A symmetric-asymmetric transition. Amplification of trap offsets due to proximity to the symmetric-asymmetric transition point.

36. A schematic illustraion: Top: initial homogeneous state. Middle: separated symmetric state. Bottom: separated asymmetric state.

37. Asymmetric states have lower interface area and energy Illustrative example: equal concentration in a cube with hard walls For the asymmetric phase, interface area is A. For the asymmetric phase, it is 3.78A Asymmetric Symmetric A

38. Different Gii’s favor the symmetric state: The state in the middle has higher density. The phase with a smaller Gii can stay in the middle to reduce the net inta-phase repulsion.

39. Physics of the interface Interface energy is of the order of in the weakly segragated regime The total density from the balance between the terms linear and quadratic in the density, the gradient term is much smaller smaller The density difference is controlled by the gradient term, however

40. Some three dimensional example

41. Broken symmetry state: Density at z=0 as a function of x and y for the TOPS trap. Right: density difference. Left: total density of 1 and 2.

42. Broken symmetry state Right: density of component 1. Left: density of component 2.

43. Symmetric state Right: density difference of 1 and 2 Left: sum of the density of 1 and 2

44. Smaller droplets: Back to symmetric state

45. Different confining potentials: The TOP magnetic trap provides for a confing potential We describe next calculations for different A/B and different densities.

46. A/B=2, Back to symmetric State

47. A/B=1.5, back to symmetric state

48. When the final phase is more symmetric: Na 2:1 Now G11>G22 Before G22>G11

49. Symmetric final State: Domain growth G11=G22 2:1

50. Amplification of the trapping potential displacement Trapping potential of the two components: dz is the displacement of one of the potential from the center. The displacement of the two components are amplified. dz

51. Expet. Result Hall et al.

52. Amplicatifation of the center of mass difference as a function of potential offset Thomas Fermi approximation: Ratio is about 70 for small offsets. For large offsets the ratio is much smaller. ``Exact calculation’’: The trend is smoother

53. Physics: Close to the critical point of change of symmetry Asymmetric solution favored by domain wall energy for G11 >G22, component 2 is inside where the density is higher and the self repulsion can be lowered. Critical point occurs when =1 In the Thomas Fermi approximation the amplification factor is proportional to 1/( - 1).

54. Boundaries of the droplet for 3% offset Nearly complete separation. Results from Thomas- Fermi approximation.

55. Density of components 1 and 2 Trap offset is only 3 per cent of the radius of the droplet. y=0 Results from Monte Carlo simulation.

56. Boundaries for 0.3% potential offset Big displacement but not yet separated. Results from Thomas- Fermi approximation.

57. Density of components 1 and 2 Trap offset is 0.3 per cent the radius of the droplet.

58. Density of component 2 Trap offset 0.3%

59. Density of component 1 Trap offset 0.3%