High Resolution Sculpting and Imaging of Ultracold Neutral Plasmas
Synthetic gauge potentials for ultracold neutral atoms ...
Transcript of Synthetic gauge potentials for ultracold neutral atoms ...
Synthetic gauge potentials for ultracold neutral atoms
Experiment (II): spin-orbit coupled Bose-Einstein condensates
ICTP, Trieste, July 6, 2012
Yu-Ju Lin
Joint Quantum Institute, National Institute of Standards and Technology, Gaithersburg and University of Maryland
Institute of Atomic and Molecular Sciences, Academia Sinica, Taiwan
Robert Compton, Karina Jimenez-Garcia, Trey Porto, William Phillips and Ian Spielman
• Optically induced vector gauge potential A* for neutral atoms:
Introduction of gauge potential
)(2
)(*
2**
xVmAqpH +
−=
***
* , ABtAE ×∇=∂
∂−=→ synthetic electric and magnetic fields
• New approach: light-induced generate B* in lab frame, no rotation of trap: (1) steady B*, not metastable (2) easy to add optical lattices
• Create synthetic field B* for neutral atoms: effective Lorentz force to simulate charged-particles in real magnetic fields
B
q
charge q
B*
q*
87Rb
B* in rotating frame: Coriolis force↔Lorentz force
Outline: synthetic gauge potentials A*
Synthetic Vector potential A*
⎥⎦
⎤⎢⎣
⎡+−= 22
**
*
2
)(2 y
xx kAqk
mH
Detuning Δ/EL
Vec
tor
pote
ntia
l q*
A* / ћ k
L
** AB ×∇=Magnetic field
Raman-dressed BEC
Raman1 Raman2
Breal
superfluid in B* (like superconductor in B)
eigenstate of “atom+photon”
Spin dependent A*: spin-orbit coupling
'↑ '↓
*↑A
*↓A
q*A*/ ћ
k↑ k↓
k0
Outline : spin-orbit coupling
• Spin-orbit (SO) coupling is important and widely studied in condensed matter physics
• Study SO coupled cold neutral atoms: clean and well controlled systems
• Some recent experiments
• Our first observation of SO coupling of neutral atoms and SO coupling of bosons: atomic COM motion and spin. Scheme: Raman laser coupling.
• Quantum phase transition due to Raman laser dressing: spatially mixed → separated Ref. Y.-J. Lin, K.J.-Garcia and Ian Spielman, Nature, Mar 3, 2011.
Introduction: spin-orbit coupling
Spin-orbit (SO) coupling: interaction between momentum k and spin σ
Rashba SO coupling in semiconductors
Simple example for electrons in solids:
iijji
jSOSO kkBH γσµ ⋅→⋅−= ∑,
)(
)(
)( 2*
xyyxSO
SO
kkH
Ekcm
kB
σσα +−=
×−=
fieldmagnetic orbit-spin effective
momentmagnetic :
:)(kBg
SO
Bσµµ =
momentum e
materials in fieldelectric static :-:
ˆ0k
zEE
=
coupling SO↔
−=∑
)(
2)(
*
*
2**
σ
i
i
ii
AmAqkH
Connection of SO coupling and spin-dependent A*
Importance of SO coupling
• Promising for making spintronics devices; spin-Hall effects Ex. Datta-Das spin field-effect transistor Ref: Datta and Das (1990); Kato. et al. (2004)
Ex.
B 0: integer quantum-Hall, chiral edge states
• Realizing topological insulators w/o breaking time reversal symmetry
Topological insulators: fermionic, insulate in the bulk, conduct in the edge states
Ex.
B=0: quantum spin-Hall, w/ SO coupling (QSH)
source (ferromagnetic)
drain (ferromagnetic)
gate
Ref: Klitzing et al. (1980) Ref: Kane and Mele, PRL (2005) Bernevig et al., Science (2006)
QH
Importance of SO coupling
• Promising for making spintronics devices; spin-Hall effects
• Realizing topological insulators w/o breaking time reversal symmetry
• w/ interaction→ topological superconductors leading to anyons, Majorana fermions, non-Abelian statistics, topological quantum computing
Ref: Qi and Zhang, Physics Today (2009) Fu and Kane, PRL (2008) Sau et al., PRL (2010) Nayak et al., Rev. Mod. Phys. (2008)
• Spin-dependent A*: non-abelian gauge potentials [ ] 0, ** ≠ji AA)σ(*iA
SO coupling for electrons
[ ])(12
22
kBBmkH SO
+⋅−⊗= µ
kinetic Zeeman SO coupling
)()(2
12
ˆ22
yyxxxyyxz kkkkmk
σσβσσασ −++−+⋅Ω
+⊗=
σi: spin ½ Pauli matrices Ω: Zeeman magnetic field along z α: Rashba SO coupling β: Dresselhaus SO coupling
α=β→ HSO=2αkxσy
),()( xySO kkkB −=
),()( yxSO kkkB −=
Rashba Dresselhaus
Electron spin helix Ref: Koralek et al., 2009.
• Hso is equivalent to that for e-, α = β
• HSO=2αkxσy term: coming from the Raman coupling Ω cos (2kLx)
SO coupling for neutral atoms
Cold neutral atoms dressed by two Raman laser beams
yxyz kmkH σασ
δσ 2
221
2
ˆ22
+⋅+⋅Ω
+⊗=
More general form:
(almost) independent control of R, D using multiple laser beams
Refs.: Campbell, Juzeliunas and Spielman, PRA (2011).
Ω,δ: effective Zeeman field along z,y HSO=2αkxσy : SO coupling
),0()( xSO kkB =
:atoms For
Raman2 Raman1
B
ωL+ΔωL ωL
Raman-dressed BEC: light-atom coupling
• Linear Zeeman shift ωZ=gµBB ≡ E-1-E0 Quadratic Zeeman shift ωq
Raman2 Raman1
B
ωL+ΔωL ωL
5S1/2
F=1 mF=0,±1
1 2
1 2
5S1/2
F=1 mF=0,±1
• Raman laser frequency difference: ΔωL Raman detuning δ= ΔωL-ωZ
ωZ ≈ΔωL ≈ 2π x 4.81 MHz
ωq ≈ 6.8 kHz
87Rb
Raman-dressed BEC: light-atom coupling
For large ωq and δ ≈0, 3-level→ effective 2-level
Raman2 Raman1
B
ωL+ΔωL ωL
⎟⎟⎠
⎞⎜⎜⎝
⎛
−Ω
Ω+⊗=
− 2/2/2/2/
12
ˆˆ2
ˆ222
δ
δxki
xki
L
L
ee
mkH
↓↑= ,basis
light-atom coupling
1 2
1 2
5S1/2
F=1 5S1/2 F=1
1 2
2 1
21ˆ2 kkek xL
−=
Ω coupling Raman :
ωZ ≈ΔωL ≈ 2π x 4.81 MHz
ωq ≈ 6.8 kHz relevant energy
⎟⎟⎠
⎞⎜⎜⎝
⎛
−−Ω
Ω+++⊗
+= Σ 2/)1(2/
2/2/)1(1
2)ˆˆ(
2
2222
δ
δ
kk
mkk
Hk
zy |↓, k-1〉 |↑,k+1〉
Raman2 Raman1
B
ωL+ΔωL ωL
1 2
2 1
quasimomentum k ↔ , labeling bare spin momentum
xk k (kL), E(EL) EL= ћ2kL
2/2m
21ˆ2 kkek xL
−=
Raman-dressed BEC: light-atom coupling
⎟⎟⎠
⎞⎜⎜⎝
⎛
−−Ω
Ω+++⊗
+= Σ 2/)1(2/
2/2/)1(1
2)ˆˆ(
2
2222
δ
δ
kk
mkk
Hk
zy |↓, k-1 〉 |↑,k+1 〉
Raman2 Raman1
B
ωL+ΔωL ωL
1 2
2 1
⎟⎟⎠
⎞⎜⎜⎝
⎛
Ω−
−Ω+++⊗=→
2/2/2/2/
ˆˆ22
12
ˆ 222
δ
δσ
ii
Ekkmm
kH LyxL
Hso =2αkxσy α=β yz σδ
σ ⋅+⋅Ω
22
Raman-dressed BEC: SO coupling
⎟⎟⎠
⎞⎜⎜⎝
⎛
−Ω
Ω+++⊗
2/2/2/2/
ˆˆ22
12
ˆ 222
δ
δσ LzxL
x Ekkmm
k
Ω: Raman coupling δ: Raman detuning
Dressed states: modified energy dispersion
• diagonalize in k space → dressed state w/ modified E(k)
⎟⎟⎠
⎞⎜⎜⎝
⎛
−−Ω
Ω+++⊗
+= Σ 2/)1(2/
2/2/)1(1
2)ˆˆ(
2
2222
δ
δ
kk
mkk
Hk
zy |↓, k-1〉 |↑,k+1〉
Raman2 Raman1
B
ωL+ΔωL ωL
1 2
2 1
quasimomentum k ↔ , labeling bare spin momentum
xk k (kL), E(EL) EL= ћ2kL
2/2m
21ˆ2 kkek xL
−=
Energy dispersion E(k) Ω= 0, δ=0
5S1/2 F=1
mF=0 mF=-1 1, +↑ k 1, −↓ k
Energy dispersion E(k) Ω= 0, 1 EL, 5 EL, δ=0
• dressed spin states
momentum quasi
near
near
=
<<Ω=
=+↑−−↓≈↓
=−↓−+↑≈↑
↓
↑
kE
kkkk
kkkk
L 18/
1,1,
1,1,'
'
ε
ε
ε
Energy dispersion at δ=0
0<Ω< 4EL: double minimum, SO coupling limit
HSO=2αkxσy > Ω
2-level, 0<Ω<5 EL, δ=0
Ω= 5 EL
Ω=0 '↓'↑
k↑ k↓
Ω=1.5 EL, δ=0
'↑ '↓
Ω> 4EL: single minimum, vector potential limit
Ref: Y.-J. Lin et al., PRL (2009).
Data: SO coupled BEC, δ=0
• Ω< 4 EL: double well regime, measure k↑, k↓ ↔ '' ↓↑ ,
• TOF images: measure quasi-momentum k
Ref. Y.-J. Lin, K.J.-Garcia and Ian Spielman, Nature, Mar 3, 2011.
0<Ω<5 EL, δ=0
k↑ k↓
k0
• Ω> 4 EL: single minimum, measure k0
'↑ '↓
• measure k↑, k↓: projection of
'' ↓↑ ,
• band mapping: easy detection of number N and temperature T of '' ↓↑ ,
↓→↓↑→↑ '' ,
Ref. Y.-J. Lin, K.J.-Garcia and Ian Spielman, Nature, Mar 3, 2011.
0 -2 2 momentum of bare spin
κ/kL
↑
↓
0 -2 2 momentum of bare spin
κ/kL
-1 +1 quasi-momentum k/kL
↑
↓
'↑ '↓
Results: double well, δ=0
0<Ω<5 EL, δ=0
k↑ k↓
k0
'↑ '↓
SO coupled BEC, δ≠ 0
E/E
L
quasi-momentum kʹ′/kL
Ω=0 δ= 0
↑ ↓
Detection: band mapping ↓→↓↑→↑ '' ,
Ω=0.19 EL th=3s
TOF x position [um]
δ= 470 Hz δ= 0
0 -260 260 0
↑
↓
↑
↓
Ω=0 δ= 0
↑
↓
TOF x position [um]
0
E/E
L
Ω=0.19 EL, hold time th=3s
δ= 0
quasi-momentum k/kL
δ>0
'↑ '↓
δ
'↑ '↓
• measure N↑’ , N↓’ at (Ω,δ) and t=th
• number is metastable up to th=3s: not ground state yet detuning width wδ decreases with th
• detuning width wδ decreases with Ω
Results: double well, δ≠0
f↓ʹ′ =0.50 ±0.16 ↔ δ=± w δ
fraction vs. δ '↓ Ω=0.6 EL th=0.1,0.5,3s Detuning width wδ vs. Ω
'↑ '↓
δ
Theory: H=Hint+ (N↑ʹ′ -N↓ʹ′)δ/2
• Hint=0, zero transition width in δ
• Hint>0: finite transition width in δ; miscible to immiscible due to Raman dressing
Experiment
• population is metastable, up to th=3s
• observe transition of miscibility
Mean field phase diagram vs. (Ω,δ)
Inset
Ground state mean-field phase diagram
Miscible to immiscible transition
Theory: T.-L. Ho (2011), Y. Li (2012) Rashba: C. Wang et al. (2010), S.-K Yip (2011).
Observing quantum phase transition: spatially mixed to separated
• phase separation • th=3s phase separation time τ
22'' ''
1↑↓↓↑−= nnnns
Ref. Y.-J. Lin, K.J.-Garcia and Ian Spielman, Nature, Mar 3, 2011.
Origin of phase separation
[ ]∫ ↓↑↓↑ ++++= ρρρρ ˆˆ)(2ˆ)(ˆ21
202
202
03
int cccccrdH
⎥⎦
⎤⎢⎣
⎡ +−+++= ↑↓↑↓↑↓∫ ρρρρρρ ˆˆ)ˆ(2
)ˆˆ)(2
(21
222222
03 ccccrd
c↑ʹ′↓ʹ′ =0, c2<0: miscible (spatially mixed)
⎥⎦
⎤⎢⎣
⎡ ++−+++=→ ↑↓↓↑↑↓↑↓∫ ''''22'
2'
22''
20
3int ˆˆ)()ˆˆ(
2)ˆˆ)(
2(
21
ρρρρρρ cccccrdH
effective interaction )8/( 220'' LEcc Ω=↓↑
Effective repulsive interaction between dressed spins c↑ʹ′↓ʹ′>0
c↑ʹ′↓ʹ′ >0
miscible, at Ω<Ωc
immiscible (spatially separated), at Ω>Ωc
in basis: )(ˆ)(ˆ)(ˆ),(ˆ)(ˆ)(ˆ '2
''2
' rerrrerr xkixki LL↑
−↓↓↓↑↑ +≈+≈ ψεψψψεψψ '' ↓↑ ,
ccccccc Ω=Ω+=++ ↓↑ at , at transition )(2 200''20
Continued work.: Williams et al., Science 335, 314-317 (2012).
Conclusion
• First experimental realization of SO coupling for neutral atoms, and for bosons
• Quantum phase transition due to Raman dressing: spatially mixed → separated
Some recent work in NIST group:
• New results (May 2012): Spin-Hall effect
• on going: Create SO coupling in fermionic 40K: towards topological insulators
Ref. Y.-J. Lin, K.J.-Garcia and Ian Spielman, Nature, Mar 3, 2011.
*'
*' ↓↑ −= BB
NIST atom chip experiment: towards topological insulators
“Raman coupling”: moving magnetic lattice
not optical, no spontaneous emission: work for alkali atoms
Theory: Goldman et al., PRL 105, 255302 (2010).
Other recent experiments
fermion
• J. Zhang group, China, arxiv 1204.1887
• M. Zwierlein group, MIT, arxiv 1205.3483
• To induce p-wave coupling between spin-polarized fermions make p-wave superfluids: non-Abelian statistics, Majorana fermions Ref: Zhang et al., PRL (2008).
boson: • S. Chen group, China, arxiv 1201.6018 collective dipole oscillations of atoms in Hso Theory: Y. Li et al. arxiv 1205.6398
Fermions with SO coupling
• (I) momentum-dependent Raman Rabi oscillation
• (II) asymmetric momentum distribution: signature of Hso
• (III) probe Fermi surface
(I) (II) (III)
40K: Wang et. al, J. Zhang, 2012. arxiv 1204.1887
22
)(2 zkm
H κσ−=
2κ
* may be useful for measuring topological invariant : probe TI and Majorana fermions
Fermions with SO coupling
Spin-orbit gap characterization Measure E(q) w/ spin texture
6Li, Cheuk et. al, M. Zwierlein, 2012. arxiv 1205.3483
|-1>, |0> mixture: c2<0, miscible for 87Rb
Ph. D thesis of M.-S. Chang, Chapman group