ming-shien chang institute of atomic and molecular sciences academia sinica dynamics of spin-1...
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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