synthetic gauge potentials for ultracold neutral atoms
TRANSCRIPT
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