Ming- Shien Chang Institute of Atomic and Molecular Sciences Academia Sinica

1. Dynamics of Spin-1 Bose-Einstein Condensates Ming-Shien Chang Institute of Atomic and Molecular Sciences Academia Sinica

2. Outline • Introduction to spinor condensates • Dynamics of spin-1 condensates • Temporal dynamics: coherent spin mixing • Spatial dynamics: miscibility and spin domain formation • Progress report: • BEC experiments at the IAMS • Summary

3. Exquisitely clean experimental system Widely variable parameters: Different atomic species Bosons, fermions Internal d.o.f. Spin systems Tunable interactions Feshbach resonances Molecular quantum gases Lattice systems Benefits from 80+yrs of theoretical many-body research Stimulating much new research Tests of mean-field theories ground state properties Interactions: repulsive, attractive, ideal gas Excitations Free expansion, vortices, surface modes Multi component mixtures Beyond mean field theories Strongly correlated systems Mott-insulator states, BCS Entanglement and squeezing Quantum Gases

4. BEC Physics BEC JILA, 1995 Order parameter χ(r) ~ N1/2ψs(r) Coherent Matter Wave Mean-field theory works

5. Phase space density • BEC occurs when: interparticle spacing, n01/3 ~ de Broglie wavelength Phase space density De Broglie wavelength Phase space density Ambient conditions 10-15 Laser cooling 10-6 Nobel Prize, 1997 BEC 1 Nobel Prize, 2001

6. Quest for BEC BEC, 1995 A. Cornell C. Wieman Nobel Prize, 2001 All-optical approach Standard recipe M. Chapman (2001) W. Ketterle (1995)

7. All-Optical BEC Gallery 1-D lattice Cross trap Single focus Common features: 87Rb CO2 trapping laser Simple MOT < 2 s evaporation time cigar disk ~ spherical 30,000 atoms 30,000 atoms 300,000 atoms

8. F = 1 mF = 1 mF = 0 mF = -1 F=1 SpinorBEC Stern-Gerlach absorption imageof a BEC created in an optical trap (GaTech, 2001)

9. Studies of F=1 Spinor BEC in an optical trap A multi-component (magnetic) quantum gas

10. Spinor Condensates A multi-component magnetic quantum gas Spinor system Spin mixing Spin domains, spin tunneling (Anti-) Ferromagnetism Rotating spinors Spin textures Skyrmion vortices Quantum Magnetism Spin squeezing, entanglement Spinors in an optical lattice Spin chains QPT, quantum quench

11. Interacting Spin-1 BEC ferromagnetic anti-ferromagnetic Intuitive picture: F = 0, 1, 2 Atomic Parameters Ho, 98 c2 << c0

12. Hamiltonian for Spin-1 BEC Ho, PRL (98) Machida, JPS (98) 2nd Quantized Form Spin changing collisions

13. Coupled Gross-Pitaevskii Eqn. for Spin-1 Condensates Condensate wave function Cross-phase modulation Modulational instability, domain formation Coherent spin (4-wave) mixing Meystre, 98-99 ……. Bigelow, 98-00

14. When c2= 0… Condensate wave function First BEC in 1995 Nobel Prize in 2001 3 Zeeman components are decoupled.

15. SpinorsIn B fields m=+1 m=0 m=-1 When linear Zeeman effects are canceled, quadratic Zeeman effect favors m0. m=+1 m=0 m=-1 72 Hz/G2 One can study spinor condensates in mG ~ G regime.

16. Single mode approximation (SMA) Simplification on spinor dynamics if all spin components have same spatial wave function (SMA): Hamiltonian reduces to just two variables to describe internal spin : Quadratic Zeeman energy Spin-dependent interaction strength Condensate magnetization Population of ±1 components follows:

17. Spinor energy contours—zero field

18. Spinor energy contours—finite field

19. Spin Mixing in spin-1 condensates t = 0 s For no interactions, m0 is lowest energy (2nd order Zeeman shift) mF = 1 0 -1 2 sec mF = 1 0 -1

20. Ferromagnetic behavior Anti-ferromagnetic spinor Ferromagnetic spinor You, 03 Chapman, 04 Sengstock, 04

21. Deterministically initiate spin mixing At t=0: (ρ1, ρ0, ρ-1) = (0, 0.75, 0.25)

22. Coherent Spin Mixing Chapman, 05 Josephson dynamics driven only by spin-dependent interactions

23. Coherent Spin Mixing Bigelow, 99 Oscillation Frequency: Direct measurement of c (c2)

24. Direct measurement of c2(or aF=2 - aF=0) aF=2 - aF=0 = -1.4(3) aB (this work) aF=2 - aF=0 = -1.40(22) aB (spect. + theory) from oscillation frequency from condensate expansion

25. Spin mixing is a nonlinear internal AC Josephson effect You, 05 de Passos, 04

26. AC Josephson Oscillations For high fields where d >> c, the system exhibits small oscillations analogous to AC-Josephson oscillations: Compare with weakly linked superconductors:

27. Controlling spinor dynamics Pulse on a magnetic field Quadratic Zeeman energy when θ (rad)

28. Controlling spinor dynamics Change trajectories by applying phase shifts via the quadratic zeeman effect Ferromagnetic ground state θ (rad)

29. Coherence of the ferromagnetic ground state Restarting the coherent spin mixing by phase-shifting out of the ground state at a later time Spin coherence time = condensate lifetime

30. Beyond the Single-Mode Approx. (SMA) Formation of spin domains Miscibilities of spin components Formation of spin waves Atomic four-wave mixing

31. Healing length shortest distance ξ over which the wavefunction can change Using Healing length: smoothes the boundary layer and determines the sizeof vortices.

32. Beyond SMA: formation of spin domains Spin healing length: weak B gradient during TOF z Single-Mode Approx. (SMA): Condensate size: (2rc,2zc) ~ (7, 70) m condensate is unstable along the z direction.

33. Miscibility of spin-1 (3-component) superfluid Goal: minimize the total mean-field energy 1-fluid M-F 2-fluid M-F 3-fluid M-F MIT, 98-99

34. Miscibility of two-component superfluids • Total Energy of two-component superfluid • If they are spatially overlapped with equal mixture: • If they are phase separated: • The condensates will phase –separated if

35. Miscibility of two-component superfluid 87Rb <1 miscible >1 immiscible 23Na >1 immiscible <1 miscible Stern-Gerlach Exp. During TOF Ferromagnetic:

36. Invalidity of the Single-Mode Approx.

37. Spin waves induced by coherent spin mixing (r1, r0, r-1) = (0, 0.75, 0.25) mF 1 0 -1 - Validate coupled GP eqn. - Theoretical explanation of spin waves. - Atomic 4-wave mixing - Evidence of dynamical instability

38. Domain formation induced by dynamical instability (r1, r0, r-1) = (0, 0.5, 0.5) mF 1 0 total -1 (r1, r0, r-1) = (0, 0.83, 0.17)

39. Miscibility of ferromagnetic spin-1 superfluid • 3 components in the ferromagnetic ground state appear to be miscible • Energy for spin waves (external) is derived from internal spinor energy mF 0 -1 1

40. Return to the SMA mF = 1 mF = 0 mF = -1 Cross trap Single focus trap

41. Validity of the SMA Condensate should be physically smaller than spin healing length Spin healing length: 1-D lattice Cross trap Single focus cigar disk ~ spherical (2rc,2zc) ~ (7, 70) m (2rc,2zc) ~ (1, 10) m (2rc,2zc) ~ (7, 7) m Condensate size

42. Improving the SMA Single-focus trap result

43. Improving the SMA Cross trap result

44. Improving the SMA

45. SMA vs. spin waves (domains) Single-focused trap Rz = 70 μm ξs = 15 μm mF 0 -1 1 Cross trap Rz = 7 μm ξs = 11 μm

46. Research projects with ultracold atoms at the IAMS • Optical dipole trap (ODT) for cold-atom experiments • Optical lattice for quantum simulation / quantum information experiment • ODT for Single atom trapping • All-optical BEC of Potassium / Rubidium • Spinor condensates studies of Potassium/ Rubidium • Determination of the spin nature of potassium • complex ground state, SSS • spin mixing of only two atoms (entangled pair after mixing) • Mixture of bosonic and fermionicspinors • Rydberg atom quantum information

47. Quest for all-Optical BEC at the IAMS

48. Optical Trap + - • Far off-resonant lasers work as static field • Focused laser beam form a 3D trap: • gaussian beam: radial • focus: longitudinal • Importance of optical trap • State-Independent Potential • Trapping of Multiple Spin States • Evaporative Cooling of Fermions

49. All-Optical BEC Gallery 1-D lattice Cross trap Single focus cigar disk ~ spherical 30,000 atoms 30,000 atoms 300,000 atoms

50. III. BEC in a Single-Focused Trap Initial loading: Scaling for Optical Trap Effective Trap Volume weak focus large trap volume low density Trap frequency Scaling for adiabatic compression Compression and evaporation: Density Elastic collision rate tight focus small trap volume high density