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. Quantum Gases Tests of mean-field theories ground state properties
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

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

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

6. Bose-Einstein Condensate:
Mean-field theory
Gross-Pitaevskii equation
(1961, nonlinear Schrodinger eqn)
s-wave scattering length

7. Quest for BEC BEC, 1995 Nobel Prize, 2001 All-optical approach
A. Cornell
C. Wieman
Nobel Prize, 2001
All-optical approach
Standard recipe
Hess, Kleppner, Greytak et al. (1986/7), Pritchard et al. (1989), Ketterle et al. (1993/4), Cornell et al. (1994)
slow (60 s)—requires exceptional vacuum not everything can be magnetically trapped magnetic fields difficult to generate
M. Chapman (2001)
W. Ketterle (1995)

8. All-Optical BEC Gallery
Cross trap
1-D lattice
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

9. F=1 Spinor BEC F = 1 mF = 1 mF = 0 mF = -1
Stern-Gerlach absorption image of a BEC created in an optical trap
(GaTech, 2001)

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

11. 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

12. Interacting Spin-1 BEC c2 << c0 Intuitive picture: F = 0, 1, 2
Atomic Parameters a0
(Bohr) a2 c0
(x10-12 Hz·cm3) c2
87Rb
101.8
100.4
7.793
23Na
50.0
55.0
15.587
0.4871
anti-ferromagnetic
ferromagnetic
Ho, 98
c2 << c0

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

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

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

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

17. 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 :
Condensate magnetization
Spin-dependent interaction strength
Quadratic Zeeman energy
Population of ±1 components follows:

18. Spinor energy contours—zero field

19. Spinor energy contours—finite field

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

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

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

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

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

25. 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

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

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

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

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

30. Search for ferromagnetic spinor ground state

31. 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

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

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

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

35. 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

36. 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

37. Miscibility of spin-1 (3-component) superfluid
2-fluid M-F
3-fluid M-F
1-fluid M-F

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

39. Invalidity of the Single-Mode Approx.

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

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

42. 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 1 -1

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

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

45. Improving the SMA
Single-focus trap result

46. Improving the SMA
Cross trap result

47. Improving the SMA

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

49. 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 fermionic spinors
Rydberg atom quantum information

50. Quest for all-Optical BEC at the IAMS

51. 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

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

53. 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

54. Dynamical Trap Compression
P = 70 w w0 70
30 μm
2.5 mm
Time
0.6 s

55. Gallery of optical lattices
In-situ imaging
CO2 lattice constant: 5.3 μm
Weiss(07)
Bookjen, PhD thesis
Time-of-flight(TOF) imaging
CO2 lattice constant: 0.43 μm
I. Bloch (01)
Greiner (09)

56. CO2 laser vs. Nd:YAG laser
 (μm)
10.6
1.06
P (W)
1000
30 (single frequency)
Scattering rate
(same trap parameters) 1
2200
Rayleigh range (same beam waist) 10
Optics
ZnSe / Ge
Usual glass
Spatial mode
Usually better
Lattice constant of an optical lattice(μm)
5.3 (easier to resolve)
0.53 (lattice physics)

57. Aspect ratio: CO2 vs. Nd:YAG

58. Trap frequency: CO2 vs. Nd:YAG

59. Trap Loading: single-focus beam
Vapor cell MOT
𝑁=2× 10 7
𝑇=30 𝜇K
Dipole trap
𝜆=1.06 μ𝑚
𝑃=8 W (each)
𝑤 0 =30 μ𝑚
𝑇 𝐷 =800 𝜇K
ω=2𝜋×(2300, 2300, 19) Hz

60. Trap Loading: cross beams
Hold time (1 sec)
𝑃=8 W (each)
x-angle = 30 ∘
𝑤 0 =30 𝜇m
𝑇 𝐷 =1.6 mK
𝜔 =2𝜋×2200 Hz

61. Free evaporation
kBT
hot atoms escape

62. Free Evaporation 𝟖 𝑾→𝟖 𝑾 𝒊𝒏 𝟏 𝒔𝒆𝒄 # of atoms N 1.0x105 trap frequency
2,120 Hz ω
13,300
rad/s
temperature T 50 μK
peak density n0
E+13
1/c.c.
phase space density Λ
8.5x10-4

63. Force evaporation 𝟖 𝑾→𝟎.𝟓 𝑾 𝒊𝒏 𝟏.𝟕 𝒔𝒆𝒄 # of atoms N 3000
trap frequency f
530 Hz ω
3300
rad/s
temperature T 6 μK
peak density n0
3.4x1011
1/c.c.
phase space density Λ
2.3x10-4

64. Spinor condensates with potassium atoms
Spinor condensates of potassium in an optical trap
Spin mixing
Determine nature of the spinors
Determine spin-dependent scattering lengths
Spinor condensates in an optical lattice
Simulation of quantum magnets
Mixture of Bosonic and Fermionic spinors

65. Zeeman slower for potassium experiment

66. Zeeman slower for potassium experiment

67. Zeeman slower for potassium experiment

68. Summary Formation of spinor condensates in all-optical traps
Observation of coherent spinor dynamics
Observation of spatial-temporal spinor dynamics
Current progress of the BEC experiments at the IAMS
Preliminary data of Rb force evaporation
Zeeman slowing of K

69. Acknowledgement
吳耿碩
陳俊嘉
黃智遠
廖冠博
鄭毓璿
彭有宏