Atomic Bose-Einstein Condensates Mixtures

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, T~Tc and 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, q2=0 when q=0. • For large enough q, q2 >0 2 q qsd

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 • N1/N2=(G12-G22)/(G11-G12) for G11>G22>G22 ( correspons to Rb) • N1/N2=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=[Ajeip(x-nl)+Bje-ip(x-nl)]ei t • Get cos(kl)=cos2qa cos2pb-(p2+q2)sin2qa sin2pb/2pq.

34. How to pick the right parameters? • k=k1+ik2; real wavevector k1 l (solid line) and imaginary wavevector k2 l (dashed line) vs 2. • Fastest mode occurs when k1 l¼

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 A Symmetric

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