Ming-Shien Chang

Institute of Atomic and Molecular SciencesAcademia Sinica

Dynamics of Spin-1 Bose-Einstein Condensates

Outline

Introduction to spinor condensates

Dynamics of spin-1 condensatesTemporal dynamics: coherent spin mixingSpatial dynamics: miscibility and spin domain formation

Progress report:BEC experiments at the IAMS

Summary

Quantum Gases

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

BEC Physics

BEC

JILA, 1995

Order parameter χ(r) ~ N1/2 ψs(r)Coherent Matter Wave

Mean-field theory works

Phase space density

BEC occurs when: interparticle spacing, n01/3 ~ de Broglie

wavelength

Phase space densityAmbient conditions 10-15

Laser cooling 10-6 Nobel Prize, 1997BEC 1 Nobel Prize, 2001

30 dBn 22 /dB Bmk T

Phase space density De Broglie wavelength

30 2.6dBn

Quest for BEC

W. Ketterle (1995)

BEC,

1995

Nobel Prize, 2001A. Cornell

C. Wieman

Standard recipe

M. Chapman (2001)

All-optical approach

All-Optical BEC Gallery

Cross trap 1-D lattice Single focus

~ spherical diskcigar

30,000 atoms 30,000 atoms 300,000 atoms

Common features:

87Rb CO2 trapping laser

Simple MOT < 2 s evaporation time

F=1 Spinor BEC

1

0

1

( )

( ) ( )

( )

r

r r

r

Stern-Gerlach absorption image of a BEC created in an optical trap(GaTech, 2001)

mF = -1

mF = 0

mF = 1

F = 1

Studies of F=1 Spinor BEC

in an optical trap

A multi-component (magnetic)

quantum gas

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

Interacting Spin-1 BEC

a0

(Bohr)

a2

(Bohr)

c0

(x10-12

Hz·cm3

)

c2

(x10-12

Hz·cm3

)

87Rb 101.8 100.4 7.793 -0.0361

23Na 50.0 55.0 15.587 0.4871 anti-ferromagnetic

ferromagnetic

Intuitive picture: F = 0, 1, 2

Atomic Parameters

20 2

0

24

3

a ac

m

22 0

2

4

3

a ac

m

c2 << c0

Ho, 98

f=1

f=1

Hamiltonian for Spin-1 BEC

3int 0 2 1 1 1 1 0 0 0 0 0

0 2 1 1 1 1

0 2 1 0 0 1 0 2 1 0 0 1

0 2 1 1 1 1

2 1 1 0 0 2 0 0 1 1

{( )

(

2( ) 2( )

2(

)

)

2 }2

c c c c

H d r c c c

c

c

c

c

c

c

Ho, PRL (98)

Machida, JPS (98)

Spin changing collisions

2nd Quantized Form

2

22 2

01

1( ) ( )

2 2( )

Ni

i i ji i

ij

j z

pH m r c r rc S S H B

m

intH

Coupled Gross-Pitaevskii Eqn. for Spin-1 Condensates

Cross-phase modulationModulational instability, domain formation

Coherent spin (4-wave) mixing

2 21,0 1,0 1 0 1( 2 ) and .L m U n n n n

1 0 1( )T

r

Condensate wave function

Bigelow, 98-00 Meystre, 98-99…….

* 211 1 0 1 2 1 0 1 1 2 1 0( )i L c n c n n n c

t

* 211 1 0 1 2 1 0 1 1 2 1 0( )i L c n c n n n c

t

*00 0 0 0 2 1 1 0 2 0 1 1( ) 2i L c n c n n c

t

When c2 = 0…

3 Zeeman components are decoupled.

First BEC in 1995Nobel Prize in 2001

2 21,0 1,0 1 0 1( 2 ) and .L m U n n n n

1 0 1( )T

r

Condensate wave function

11 1 0 1i L c n

t

11 1 0 1i L c n

t

00 0 0 0i L c n

t

Spinors In B fields

20 1 12 c n

Bm m m

72 Hz/G2

One can study spinor condensates in mG ~ G regime.

When linear Zeeman effects are canceled,

quadratic Zeeman effect favors m0.

2 10 Hz,c n

m=+1 m=0 m=-1

m=+1 m=0 m=-1

0 22

272

E E EB

Single mode approximation (SMA)

4 32 2( )c c r d r c n +1 1 0=(E 2 ) / 2E E 1 1M

Spin-dependent interaction strength

Quadratic Zeeman energy

Condensate magnetization

Hamiltonian reduces to just two variables to describe internal spin :

0( ) 2t

20 0 0

20[(1 (1 ) cos )] (1 )E c M

1

0

1

1 1

0 0

1 1

( )

( ) ( ) ( )

( )

i

i

i

r e

r r n r e

r e

Simplification on spinor dynamics if all spin components have same spatial wave function

(SMA):

1 0(1 ) / 2M Population of ±1 components follows:

0 0 1 0 1(t)= /( ),n n n n

Spinor energy contours—zero field

2 20 0 0 0[(1 ) (1 ) cos ] (1 ),E c M

1.0

0.8

0.6

0.4

0.2

0.0

-6 -4 -2 0 2 4 6

Ferromagnetic1.0

0.8

0.6

0.4

0.2

0.0

-6 -4 -2 0 2 4 6

Anti-ferromagnetic

Spinor energy contours—finite field

2 20 0 0 0[(1 (1 ) cos )] (1 ),E c M

1.0

0.8

0.6

0.4

0.2

0.0

-6 -4 -2 0 2 4 6

Ferromagnetic1.0

0.8

0.6

0.4

0.2

0.0

-6 -4 -2 0 2 4 6

Anit-ferromagnetic

Spin Mixing in spin-1 condensates

2 sec

For no interactions,

m0 is lowest energy

(2nd order Zeeman shift)

20 1 12 c n

Bm m m

mF = 1 0 -1

mF = 1 0 -1

t = 0 s

Ferromagnetic behavior

Ferromagnetic spinor

Anti-ferromagnetic spinor

Chapman, 04 You, 03

Sengstock, 04

Deterministically initiate spin mixing

At t=0: (ρ1, ρ0, ρ-1) = (0, 0.75, 0.25)

. and )2

( where 1010,1

22

0,1

nnnnUm

L

* 211 1 0 1 2 1 0 1 1 2 1 0( )i L c n c n n n c

t

*00 0 0 0 2 1 1 0 2 0 1 1( ) 2i L c n c n n c

t

* 211 1 0 1 2 1 0 1 1 2 1 0( )i L c n c n n n c

t

Coherent Spin Mixing

Chapman, 05

Josephson dynamics driven only by spin-dependent interactions

Coherent Spin Mixing

1 1 0( 2 )c Oscillation Frequency: Bigelow, 99

Direct measurement of c (c2)

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)

/ 2 4.3(3) rad/sc

14 30 2.1(4) 10 cmn

4 32 2 0

4( )

7c c N r d r c n

from oscillation frequency

from condensate expansion

4 32 2where ( ) ,c c N r d r c n

20 0

0 2 20

(1 )(1 2 )2 2 2(1 2 ) cos

(1 )

Mc c

M

2 20 0 0

2(1 ) sin

cM

2+1 1 0=(E 2 ) / 2 2 72 ,E E B

Spin mixing is a nonlinear internal AC Josephson effect

1 -1= - ,M You, 05

de Passos, 04

20 0 0

20[(1 (1 ) cos )] (1 )E c M

0 0 1 0 1(t)= /( ),n n n n 1 1 0( ) 2 ,t

AC Josephson Oscillations

0 ( ) ( / )sint A

( ) 2 /t

For high fields where d >> c, the system exhibits small oscillations analogous to AC-Josephson oscillations:

( ) sincI t I

( ) 2 /t eV

Compare with weakly linked superconductors:

Controlling spinor dynamics

+1 1 0=(E 2 ) / 2E E

Quadratic Zeeman energy

0.5

0.4

0.3

0.2

0.1

0.0

(rad)θ (rad)

(2 / ) dt

2d

dt

when 2c c n

Pulse on a magnetic field

Controlling spinor dynamics

(2 / ) dt

Change trajectories by applying phase shifts via the quadratic zeeman effect

2

t

θ (rad)

Ferromagnetic ground state

Coherence of the ferromagnetic ground state

0.5

0.4

0.3

0.2

0.50.0B (G

)

0.60.40.20.0Time (s)

Restarting the coherent spin mixing by phase-shifting out of the ground state at a later time

Spin coherence time = condensate lifetime

Beyond the Single-Mode Approx. (SMA)

Formation of spin domains

Miscibilities of spin components

Formation of spin waves

Atomic four-wave mixing

Healing length

Healing length: smoothes the boundary layer and determines the size of vortices.

2 22

22 2M M

0 /n gUsing

shortest distance ξ over which the wavefunction can change

2 2

0 02

4

2

an n

M M

g

01/ 8 an

Beyond SMA: formation of spin domains

2

~ 15 m2

smc n

Spin healing length:

Condensate size: (2rc,2zc) ~ (7, 70) m

condensate is unstable along the z direction.2 > :c sz

Single-Mode Approx. (SMA): ( ) ( ) jij jr r e

weak B gradient during TOF

z

Miscibility of spin-1 (3-component) superfluid

3 2 2 21 1 0 0

10 1 0 10 1 0 1

00,1 1 0 1

1

1

1 1

1

1

2

1

4

{2

2 2

}

MF

g n n g n n g n

E d r g n g n g n

n

n

g n n

1-fluid M-F

2-fluid M-F

3-fluid M-F

Goal: minimize the total mean-field energy

0 2 0 20 2

0 2 0 20

0 2 0 20 2

1 0 1

12 2

02 2

12 2

ijg m m m

c c c cm c c

c c c cm c

c c c cm c c

00,1 1 2g c

MIT, 98-99

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

22 2 31 4

( 2 ) , g2 a a b b ab a b

aE g n g n g n n d r

m

)2(2

2

abba gggV

NEo

)(2

2

b

b

a

aS V

g

V

gNE

ab a bg g g

Miscibility of two-component superfluid

201,1

200,1

00

2011

ccg

ccg

cg

ccgg

<1miscible

>1immiscible23Na

>1immiscible

<1miscible87Rb

1,0

1 0

g

g g

1, 1

1 1

g

g g

2 0c Ferromagnetic:

Stern-Gerlach Exp. During TOF

Invalidity of the Single-Mode Approx.

80-80Rz (m)

0.8

0.6

0.4

0.2

0.0

Tim

e (s

)

Spin waves induced by coherent spin mixing

0.8

0.6

0.4

0.2

0.0

mix

ing

time

(sec

)

mF 1 0 -1

- Validate coupled GP eqn.- Theoretical explanation of

spin waves.- Atomic 4-wave mixing- Evidence of dynamical

instability

(r1, r0, r-1) = (0, 0.75, 0.25)

0.8

0.6

0.4

0.2

0.0

Tim

e (s

)

80-80Rz (m)

0.6

0.5

0.4

0.3

0.2

0.1

0.0

Tim

e (s

)

mF 1 0 -1

(r1, r0, r-1) = (0, 0.5, 0.5)

(r1, r0, r-1) = (0, 0.83, 0.17)

total

Domain formation induced by dynamical instability

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

0.6

0.5

0.4

0.3

0.2

0.1

0.0

Tim

e (s

)

(c)

1

2

mF 1 0 -1

Return to the SMA

mF = -1

mF = 0

mF = 1

Single focus trap Cross trap

Validity of the SMA

2

~ 10 15 m2

s

h

mc n Spin healing length:

(2rc,2zc) ~ (7, 70) m

Condensate should be physically smaller than spin healing length

Cross trap 1-D lattice Single focus

~ spherical diskcigar

(2rc,2zc) ~ (1, 10) m

Condensate size

(2rc,2zc) ~ (7, 7) m

Improving the SMA

1.0

0.8

0.6

0.4

0.2

0.0

m

0.80.60.40.20.0

Time (s)

Single-focus trap result

Improving the SMA

1.0

0.8

0.6

0.4

0.2

0.0

m

1.21.00.80.60.40.20.0

Time (s)

Cross trap result

Improving the SMA

1.0

0.8

0.6

0.4

0.2

0.0

m

1.21.00.80.60.40.20.0

Time (s)

1.0

0.8

0.6

0.4

0.2

0.0

m

1.21.00.80.60.40.20.0

Time (s)

SMA vs. spin waves (domains)0.6

0.5

0.4

0.3

0.2

0.1

0.0

Tim

e (s

)(b)

1.2

1.0

0.8

0.6

0.4

0.2

0.0

Tim

e (s

)

(d)

-80 0 80Rz (m)

Single-focused trap

Rz = 70 μm

ξs = 15 μm

Cross trap

Rz = 7 μm

ξs = 11 μm

mF 1 0 -1

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

Quest for all-Optical BEC at the IAMS

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

20

2

20

/2

/11

0r

zzeUU

2

0

1

2 2U E I

c

+-

All-Optical BEC Gallery

Cross trap 1-D lattice Single focus

~ spherical diskcigar

30,000 atoms 30,000 atoms 300,000 atoms

III. BEC in a Single-Focused Trap

weak focuslarge trap volume

low density

Initial loading:

tight focussmall trap volume

high density

Compression and evaporation:

Scaling for Optical Trap

Scaling for adiabatic compression

Density

Elastic collision rate

nw0 4

50

21 wP

Effective Trap Volume

Trap frequency

40wV

370

21 wP

Dynamical Trap Compression

Time 0 0.6 s

2.5

mm

P = 70 ww0 30 μm70

Gallery of optical lattices

In-situ imagingCO2 lattice constant: 5.3 μm

Time-of-flight(TOF) imagingCO2 lattice constant: 0.43 μm

Bookjen, PhD thesis

I. Bloch (01)

Weiss(07)

Greiner (09)

CO2 laser vs. Nd:YAG laser

CO2 laser Nd:YAG

(μm) 10.6 1.06

P (W) 1000 100030 (single frequency)

Scattering rate (same trap parameters)

1 2200

Rayleigh range (same beam waist)

1 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)

1

2

3

4

56

10

2

3

4

56

100

Asp

ect

Rat

io

9080706050403020100Crossing Angle (degree)

fx / fz (YAG) fy / fz (YAG)

fx / fz (CO2) fy / fz (CO2)

Aspect ratio: CO2 vs. Nd:YAG

2500

2000

1500

1000

trap

fre

quen

cy (

Hz)

9080706050403020100crossing angle (degree)

1.0

0.9

0.8

0.7

0.6

0.5

0.4

fYA

G / fC

O2

0.986

f (YAG) f (CO2) f_YAG / f_CO2

Trap frequency: CO2 vs. Nd:YAG

Trap Loading: single-focus beam

𝑤0=30 μ𝑚

(each)

𝜆=1.06 μ𝑚

Vapor cell MOT

𝑁=2×107

Dipole trap

𝑇 𝐷=800𝜇K

ω=2𝜋×(2300 ,23 00 ,19)Hz

𝑇=30𝜇 K

Trap Loading: cross beams

Hold time (1 sec)

𝑤0=30𝜇m

(each)

𝜆=1.06𝜇m

x-angle

𝑇 𝐷=1.6mK

Free evaporation

kBT

hot atoms escape 3 30 ( / )dB Bn N k T

Free Evaporation

# of atoms N 1.0x105

trap frequency f 2,120 Hz

trap frequency ω 13,300 rad/s

temperature T 50 μK

peak density n0 4.57147E+13 1/c.c.

phase space density Λ 8.5x10-4

𝟖𝑾→𝟖𝑾 𝒊𝒏𝟏 𝒔𝒆𝒄

Force evaporation

# of atoms N 3000

trap frequency f 530 Hz

trap frequency ω 3300 rad/s

temperature T 6 μK

peak density n0 3.4x1011 1/c.c.

phase space density Λ 2.3x10-4

𝟖𝑾→𝟎 .𝟓𝑾 𝒊𝒏𝟏 .𝟕𝒔𝒆𝒄

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

Zeeman slower for potassium experiment

Zeeman slower for potassium experiment

Zeeman slower for potassium experiment

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 IAMSPreliminary data of Rb force evaporationZeeman slowing of K

Acknowledgement

吳耿碩陳俊嘉黃智遠廖冠博鄭毓璿彭有宏