Mueller, et al., PRA 70, 041603 (2004)

A.S. Sorensen et al., PRL 94, 086803 (2005).

g e

two internal state

D. Jaksch and P. Zoller, New J. Phys. 5, 56 (2003)

H ˆ = - t

_{j,j ¢}

2 ( a ˆ

^†_j

a ˆ

_{j ¢}

e

^{-iAj,j ¢}

+ h.c. )

j,j ¢

å ^{+ E}

^j

^{N ˆ}

^j

å

j

^{+ U} ₂

^j

^{N ˆ}

^j

( ^{N ˆ}

^j

^{- 1} )

å

j

Peierls phase

光格子中の原子系における

Hofstadter-Harper

ハミルトニアンの実現

(Humburg) J. Struck, PRL 108, 225304 (2012), arXiv:1304.5520

(MIT) H. Miyake, arXiv:1308.1431

(MaxPlanck) M. Aidelsburger, PRL 107, 225301 (2011), arXiv:1308.0321

スピン軌道相互作用

_(SOC)

B

_SO

=

mc

²

( E ´ k ) ⁼ _mc

2

E

₀

( - k

_y

, k

_x

,0 )

粒子が動く座標系における磁場と粒子の 磁気モーメントとの相互作用。

x z

+ + +

− − −

+ + +

− − − J B

J k =

E = E

₀

z ˆ

^のとき

Rashba SOC

２次元電子系（

GaAs

系，

etc

）

Dresselhaus SOC

半導体の結晶構造に起因

→

スピンホール効果，トポロジカル絶縁体，

etc

y

SOC

をもつボソン系

Rashba

型相互作用



H  h

²

2m  k

²

- 2   k

_x



_x

 k

_y



_y

 

3 nents

U↑λ(χp) =

cos²χp + ∆²sin²χp − ∆ λsinχp 1/2

√2 cos²χp + ∆²sin²χp

1/4 and

(5) U_↓λ(χp) = − iλ sign [cosχp] U_↑−λ(χp), (6) whereχp is the azimuthal angle in the (px, py)-plane and

∆ = v /v < 1. T he unitary matrix U_{α λ}(χp) diagonalizes the H am iltonian (1) (where α = ↑,↓corresponds to the pseudo-spin index and λ = ±1 labels the eigenstates).

I t is obvious from E q. (4) that the spectrum of the sin-gle particle problem contains two m inima at λ = − 1 and m om enta p_y = p_z = 0 and p_x = ±m v = 0 (see F ig. 1). Consequently, the single particle ground-state is double-degenerate and the m ost general expression for the corresponding wave-function is

Ψd w(r ) = √

wL 1

− i e− i m v x + iφL+√

wR 1

i ei m v x + iφR , (7) where wL ≥ 0 and wR ≥ 0 are the fractions of

“left-” and “right-m oving“left-” states sub jected to the constraint wL+ wR = 1, whileφL and φR are arbitrary phases. N ote that by left/right-m oving states we m ean states with non-zero m om entum average, p = ∓ m vex. H owever, the corresponding average velocity vanishes ∇pH (p ) = 0,ˇ so that quasiparticles characterized by these non-zero m om entum single-particle states are not actually “m ov-ing”, as long as the laser fi elds generating the spin-orbit coupling are m aintained. N ote that rotations in the m an-ifold of the double-well ground-states are distinct from rotations in the pseudo-spin H ilbert space, as real-space and pseudo-spin coordinates are mixed up by the spin-orbit interaction. T he two-fold degeneracy of the single-particle ground state is preserved if the system is placed in a harm onic trap. For a potential Vtra p = mω²r²/2, we can write the Sch¨odinger equation in mom entum rep-resentation: T he trap potential plays the role of “the kinetic energy” and the real kinetic term produces a double-well potential in m omentum space, see F ig. 1.

T he tunnelling processes connect the degenerate vacua in m om entum space²⁴. H owever, they do not elim inate the double-degeneracy of the single-particle states, which is protected by the K ram ers-like symm etry (see Section I I I B).

At low tem peratures, the m any-body Bose system (1) condenses into the single-particle states corresponding to the double-well minim a. T he transition tem perature of this double-well SOBE C can be calculated using stan-dard text-book procedures.²⁵ Let us assum e that near and below the transition the band with λ = + 1 does not contribute and that we can expand the spectrum near the m inim a of the band (4). W e defi ne the mom en-tum q in the vicinity of the left/right m inim a as follows:

p = ±m vex + q , with q m v. E q. (4) leads to the

P

x y

P E(p)

F I G . 2: (color online): Schem atic picture of the band struc-ture describ ed by E q. (4) for the isotropic R ashba-typ e case with v = v for pz = 0. T he inside sheet represents theλ = + 1 band, while the outside sheet corresp onds to λ = − 1 and has m inim a a one-dim ensional circle p

p²_x + p²_y = m v.

anisotropic spectrum : δE (q ) = q²_x + q_z²

2m + 1 − v

v

2 q²_y

2m. (8)

T he transition tem perature is Tc = π

2 4 ζ(3/2)

3 2

1 − v v

2 ¹3

n²³

m . (9)

W e see that if n^1/3 1 − (v /v)^{2 1/6} m v, our approxi-m ation is justifi ed and, in particular, the density of par-ticles in the upper band λ = + 1 is exponentially sm all.

I n the isotropic limit ∆ = v /v → 1, the transition tem perature form ally vanishes. N ote that in the isotropic case v = v the spin-orbit term of the H am iltonian (1) is equivalent to the R ashba m odel²⁶ and can be reduced to the latter via the rotation exp (iπˇσ2/4) in the pseudo-spin space. I n this case, the spectrum (4) has m inima on a one-dimensional circle p²_x + p²_y = m v (see F ig. 2).

T he single-particle ground-state is infi nitely degenerate and the most general expression for the corresponding wave-function is

Ψri n g(r ) =

2π

0

dχ

2π w(χ) U−(χ)e^iφ(χ)e[i m v ( x cosχ+ y si nχ) ], (10) where w(χ) > 0 is the angle-dependent weight of the Bose-condensate on a circle [ dχ/(2π)w(χ) = 1] and φ(χ) is the angle-dependent phase. A n especially in-teresting class of ground states corresponds to w(χ) not

Single-particle dispersion

Stanescu, et al., PRA 78, 023616 (2008)

k‖−S

k‖S

Minima at k

_⊥

=

±

 Degenerate ground state for different azimuthal angle

・ 最低エネルギー状態が縮退。 相互作用のある系での凝縮状態の構造？

縮退は原子間相互作用、トラップの非等方性、、、などにより解ける。

冷却原子系における スピン軌道相互作用

Raman coupling

Lin, et al., Nature 471, 83 (2011)

Hamiltonian

Rashba+Dresselhaus type vector potential



H

_eff⁽²⁾

 - h

²



²

2m 1 ˆ    ^ˆ

_z

 0  ˜ e

^-2ikLx

 ˜ e

^2ikLx

0



  

 

¢

H

_eff^{(2 )}

= U ˆ

^†

( ) r ^H

^eff(2)

^{U ˆ} ( ) ^r

= -

²

2 m ( iÑ - k

_L

s ˆ

_x

^{x ˆ} )

²

^{- d s ˆ}

^x

^{+W s ˆ}

^z

Pseudospin representation

- = e

^{-ip 4}

{ cos ( ) k

_L

x ^{1 + sin} ( ) ^k

^L

^{x 2} }

¯ = e

^{ip 4}

{ - sin ( ) k

_L

x ^{1 + cos} ( ) ^k

^L

^{x 2} }



U ˆ (r)  e

^{-iˆ z 4}

e

^{ikLxˆ y}

Unitary transformation

(Non-Abelian gauge field)

冷却原子系における スピン軌道相互作用

Boson

Fermion

87 Rb F=1 m

_F

= 1, 0, × - 1

Lin et al., Nature 471, 83 (2011)

40 K F=9/2 m

_F

= 9 2, 7 2

Wang et al., PRL 109, 095301 (2012)

6 Li F=3/2 m

_F

= 3 2, 1 2

Rashba+Dresselhaus type vector potential

¢

H

_eff^{(2 )}

= U ˆ

^†

( ) r ^H

^eff(2)

^{U ˆ} ( ) ^r

= -

²

2 m ( iÑ - k

_L

s ˆ

_x

^{x ˆ} )

²

^{- d s ˆ}

^x

^{+W s ˆ}

^z

(Non-Abelian gauge field)

Raman coupling

様々なスピン軌道相互作用 の生成法の提案

Lin et al., Nature 471, 83 (2011)

s

_x

^{x ˆ}

Rashba

+Dresselhause (1D)

s

_x

^{x ˆ} + s

_y

^{y ˆ}

s

_x

^{x ˆ} + s

_y

^{y ˆ} + s

_z

^{z ˆ}

Rashba (2D)

Rashba (3D)

Campbell et al., PRA. 84, 025602 (2011)

Juzeliūnas et al., PRA 81, 053403 (2010)

Anderson et al., PRL 108, 235301 (2012)

Xu et al., RPA 87, 063634 (2013)

Anderson et al., arXiv:1306.2606

磁場勾配を用いた提案

冷却原子系におけるスピン軌道 相互作用の実験的観測

Lin et al., Nature 471, 83 (2011)

H ¢

_eff⁽²⁾

= -

²

2 m ( iÑ - k

_L

s ˆ

_x

^{x ˆ} )

²

^{- d s ˆ}

^x

^{+W s ˆ}

^z

分散関係

TOF imaging

スピン軌道相互作用をもつ

_BEC

基底状態 ボース凝縮

分散関係

Rashba+Dresselhaus SOC BEC

E

_int

= ò d r ^æ _è ^{ç g +} ₂ ^g

¹²

ⁿ

²

^{+ g -} ₂ ^g

¹²

^S

^z2

^ö _ø ^÷



g  g

₁₁

 g

₂₂

g

₁₂

 g  

_R

2 E

_L

Lin et al., Nature 471, 83 (2011)

= 0.1 EL = 0.3 EL = 0.6 EL

10^-1

10^-2

0

-10^-2

-10^-1 Detuning/EL

0.6 0.5

0.4 0.3

0.2 0.1

0.0

Raman Coupling /EL

Mixed

Metastable window Phase

Separated -4

-2 0 2 4

Detuning/EL

5 4

3 2

1 0

Raman Coupling /E_L

Single minimum

Inset

Equal population a Mean field phase diagram

c Miscible to immiscible transition

Phase Mixed Phase Separated Raman

Coupling /E_L

0.0 0.19

b Phase diagram, inset

Figure 2 | Phases of a SO coupled BEC. a-b, Mean field phase diag rams for infinite homo-geneous SO coupled ⁸⁷Rb BE Cs (1.5 kHz chemical potential). The background colors indicate atom fraction in |" i and |#i. Between the dashed lines there are two dressed spin states, |"⁰i and |#⁰i. a, Single particle phase diag ram in the ⌦− δ plane. b, Phase diag ram as modified by interactions. The dots represent a metastable region where the fraction of atoms f_"⁰_,#⁰ remains largely unchanged for t_h = 3 s. c, Phase line for mixtures of dressed spins and images after TOF (with populations N_" ⇡ N_#), mapped from |"⁰i and |#⁰i showing the transition from phase-mixed to phase-separated within the “metastable window” of detuning.

23

= 0.1 E_L = 0.3 E_L = 0.6 E_L

10^-1

10^-2

0

-10^-2

-10^-1 Detuning/EL

0.6 0.5

0.4 0.3

0.2 0.1

0.0

Raman Coupling /E_L Mixed

Metastable window Phase

Separated -4

-2 0 2 4

Detuning/EL

5 4

3 2

1 0

Raman Coupling /E_L

Single minimum

Inset

Equal population a Mean field phase diagram

c Miscible to immiscible transition

Phase Mixed Phase Separated Raman

Coupling /E_L

0.0 0.19

b Phase diagram, inset

Figure 2 | Phases of a SO coupled BEC. a-b, Mean field phase diagrams for infinite homo-geneous SO coupled

⁸⁷

Rb BE Cs (1.5 kHz chemical potential). The background colors indicate atom fraction in |" i and |#i . Between the dashed lines there are two dressed spin states, |"

⁰

i and |#

⁰

i . a , Single particle phase diagram in the ⌦− δ plane. b , Phase diagram as modified by interactions. The dots represent a metastable region where the fraction of atoms f

_"⁰_,#⁰

remains largely unchanged for t

_h

= 3 s. c , Phase line for mixtures of dressed spins and images after TOF (with populations N

_"

⇡ N

_#

), mapped from |"

⁰

i and |#

⁰

i showing the transition from phase-mixed to phase-separated within the “metastable window” of detuning.

23



H  dr

_^

h

²

2m  -i - k

_L

 ˆ

_x

x ^ˆ 

²

^{-   ˆ}

x

  ˜  ˆ

_z



  

 

 ^

^

^{ 1} ₂  ^drn

^

^g

^

ⁿ

^



n  n

₁

 n

₂

S

_z

 n

₁

- n

₂

Rashba+Dresselhaus SOC BEC

Lin et al., Nature 471, 83 (2011)

= 0.1 E_L = 0.3 E_L = 0.6 E_L

10^-1

10^-2

0

-10^-2

-10^-1 Detuning/EL

0.6 0.5

0.4 0.3

0.2 0.1

0.0

Raman Coupling /E_L Mixed

Metastable window Phase

Separated -4

-2 0 2 4

Detuning/EL

5 4

3 2

1 0

Raman Coupling /E_L

Single minimum

Inset

Equal population a Mean field phase diagram

c Miscible to immiscible transition

Phase Mixed Phase Separated Raman

Coupling /E_L

0.0 0.19

b Phase diagram, inset

Figure 2 | Phases of a SO coupled BEC. a-b, Mean field phase diagrams for infinite homo-geneous SO coupled

⁸⁷

Rb BE Cs (1.5 kHz chemical potential). The background colors indicate atom fraction in |" i and |#i . Between the dashed lines there are two dressed spin states, |"

⁰

i and |#

⁰

i . a , Single particle phase diagram in the ⌦− δ plane. b , Phase diagram as modified by interactions. The dots represent a metastable region where the fraction of atoms f

_"⁰_,#⁰

remains largely unchanged for t

_h

= 3 s. c , Phase line for mixtures of dressed spins and images after TOF (with populations N

_"

⇡ N

_#

), mapped from |"

⁰

i and |#

⁰

i showing the transition from phase-mixed to phase-separated within the “metastable window” of detuning.

23

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