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