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ISTITUTO NAZIONALE

DI OTTICA

INO

UNIVERSITA DEGLI STUDI

DI TRENTO

Superstripes and the excitation spectrum of

a spin-orbit-coupled BEC

Yun Li, Giovanni I. Martone, Lev P. Pitaevskii,and Sandro Stringari

Trieste, 2013 May

Breaking of two symmetries

Bose condensation

of defectons

A. F. Andreev and I. M. Lifshitz

JETP 29, 1107 (1969)

Yun Li, Giovanni I. Martone, Lev P. Pitaevskii, and Sandro Stringari Superstripes and the excitation spectrum of a spin-orbit-coupled BEC

Breaking of two symmetries

Bose condensation

of defectons

A. F. Andreev and I. M. Lifshitz

JETP 29, 1107 (1969)

Soft-core interactions

0 0.5 1 1.5 2

x

0

0.5

1

1.5

y

0 0.5 1 1.5 2

x

0

0.5

1

1.5

y

(a)

(c) (d)

(b)

N. Henkel, et al., PRL 104, 195302 (2010)

F. Cinti, et al., PRL 105, 135301 (2010)

Yun Li, Giovanni I. Martone, Lev P. Pitaevskii, and Sandro Stringari Superstripes and the excitation spectrum of a spin-orbit-coupled BEC

Spin-orbit coupled BEC (single particle picture)

bulk, no interactions, Ω = 0

−2 −1 0 1 2

−0.5

0

0.5

1

p / k0

E(p

) / k

02 |↓⟩ |↑⟩

BEC

Single-particle Hamiltonian

h0 =1

2

ˆ

(px − k0σz)2 + p2

˜

2σx +

δ

2σz

Equal Rashba and Dresselhauscouplings

⇐= U−1 hlab0 U

U = eiΘ(x)σz/2

Yun Li, Giovanni I. Martone, Lev P. Pitaevskii, and Sandro Stringari Superstripes and the excitation spectrum of a spin-orbit-coupled BEC

Spin-orbit coupled BEC (single particle picture)

bulk, no interactions, Ω 6= 0

−2 −1 0 1 2

−0.5

0

0.5

1

p / k0

E(p

) / k

02 |↓⟩ |↑⟩|↓⟩|↑⟩

BEC

Single-particle Hamiltonian

h0 =1

2

ˆ

(px − k0σz)2 + p2

˜

2σx +

δ

2σz

Equal Rashba and Dresselhauscouplings

⇐= U−1 hlab0 U

U = eiΘ(x)σz/2

±k1 = ±k0

s

1 − Ω2

4k40

Yun Li, Giovanni I. Martone, Lev P. Pitaevskii, and Sandro Stringari Superstripes and the excitation spectrum of a spin-orbit-coupled BEC

Spin-orbit coupled BEC (single particle picture)

bulk, no interactions, Ω 6= 0

−2 −1 0 1 2

−0.5

0

0.5

1

p / k0

E(p

) / k

02 |↓⟩ |↑⟩|↓⟩|↑⟩

BEC

Single-particle Hamiltonian

h0 =1

2

ˆ

(px − k0σz)2 + p2

˜

2σx +

δ

2σz

Equal Rashba and Dresselhauscouplings

⇐= U−1 hlab0 U

U = eiΘ(x)σz/2

±k1 = ±k0

s

1 − Ω2

4k40

Yun Li, Giovanni I. Martone, Lev P. Pitaevskii, and Sandro Stringari Superstripes and the excitation spectrum of a spin-orbit-coupled BEC

Spin-orbit coupled BEC (single particle picture)

bulk, no interactions, Ω 6= 0

−2 −1 0 1 2

−0.5

0

0.5

1

p / k0

E(p

) / k

02 |↓⟩ |↑⟩|↓⟩|↑⟩

BEC

Single-particle Hamiltonian

h0 =1

2

ˆ

(px − k0σz)2 + p2

˜

2σx +

δ

2σz

Equal Rashba and Dresselhauscouplings

⇐= U−1 hlab0 U

U = eiΘ(x)σz/2

±k1 = ±k0

s

1 − Ω2

4k40

Yun Li, Giovanni I. Martone, Lev P. Pitaevskii, and Sandro Stringari Superstripes and the excitation spectrum of a spin-orbit-coupled BEC

Spin-orbit coupled BEC (single particle picture)

bulk, no interactions, Ω 6= 0

−2 −1 0 1 2

−0.5

0

0.5

1

p / k0

E(p

) / k

02 |↓⟩ |↑⟩|↓⟩|↑⟩

BEC

Single-particle Hamiltonian

h0 =1

2

ˆ

(px − k0σz)2 + p2

˜

2σx +

δ

2σz

Equal Rashba and Dresselhauscouplings

⇐= U−1 hlab0 U

U = eiΘ(x)σz/2

±k1 = ±k0

s

1 − Ω2

4k40

Yun Li, Giovanni I. Martone, Lev P. Pitaevskii, and Sandro Stringari Superstripes and the excitation spectrum of a spin-orbit-coupled BEC

Spin-orbit coupled BEC (single particle picture)

bulk, no interactions, Ω 6= 0

−2 −1 0 1 2

−0.5

0

0.5

1

p / k0

E(p

) / k

02 |↓⟩ |↑⟩|↓⟩|↑⟩

BEC

Single-particle Hamiltonian

h0 =1

2

ˆ

(px − k0σz)2 + p2

˜

2σx +

δ

2σz

Equal Rashba and Dresselhauscouplings

⇐= U−1 hlab0 U

U = eiΘ(x)σz/2

±k1 = ±k0

s

1 − Ω2

4k40

Yun Li, Giovanni I. Martone, Lev P. Pitaevskii, and Sandro Stringari Superstripes and the excitation spectrum of a spin-orbit-coupled BEC

Spin-orbit coupled BEC (single particle picture)

bulk, no interactions, Ω 6= 0

−2 −1 0 1 2

−0.5

0

0.5

1

p / k0

E(p

) / k

02 |↓⟩ |↑⟩|↓⟩|↑⟩

BEC

Single-particle Hamiltonian

h0 =1

2

ˆ

(px − k0σz)2 + p2

˜

2σx +

δ

2σz

Equal Rashba and Dresselhauscouplings

7

6

5

4

3

2

1

0

gnil

pu

oC

na

ma

R/EL

-1.0 -0.5 0.0 0.5 1.0

Minima location in units of kL

±k1 = ±k0

s

1 − Ω2

4k40

Lin et al, Nature

83, 1523 (2010)

Yun Li, Giovanni I. Martone, Lev P. Pitaevskii, and Sandro Stringari Superstripes and the excitation spectrum of a spin-orbit-coupled BEC

Many-body ground state

Three quantum phases γ = (g − g↑↓) / (g + g↑↓) > 0 , n(c) = k20/ (2γg)

(I). k1 6= 0, C+ = C−,〈σz〉 = 0

(II). k1 6= 0, C− = 0 orC+ = 0, 〈σz〉 6= 0

(III). k1 = 0, 〈σz〉 = 0

ψ =

ψ↑

ψ↓

«

=√n

»

C+

cos θ− sin θ

«

eik1x +C−

sin θ− cos θ

«

e−ik1x

cos 2θ =k1

k0, 〈σz〉 =

k1

k0

`

|C+|2 − |C−|2´

LY, Pitaevskii, Stringari, PRL 108, 225301 (2012)

Yun Li, Giovanni I. Martone, Lev P. Pitaevskii, and Sandro Stringari Superstripes and the excitation spectrum of a spin-orbit-coupled BEC

Many-body ground state

Three quantum phases γ = (g − g↑↓) / (g + g↑↓) > 0 , n(c) = k20/ (2γg)

(I). k1 6= 0, C+ = C−,〈σz〉 = 0

(II). k1 6= 0, C− = 0 orC+ = 0, 〈σz〉 6= 0

(III). k1 = 0, 〈σz〉 = 0

Ω(I-II)cr = 2k2

0

p

2γ/(1 + 2γ) small for 87Rb

Ω(II-III)cr = 2k2

0

1.0

0.5

0.03210

Hold time th (s)

= 0.14(3) s

Dynamics at = 0.6 EL

1.0

0.8

0.6

0.4

0.2

0.0

,n

oitar

ap

es

es

ah

Ps

0.012 3 4 5 6 7

0.12 3 4 5 6 7

12

Raman Coupling /EL

C = 0.20(2) EL

Ω

Ω

τ

Ω

Yun Li, Giovanni I. Martone, Lev P. Pitaevskii, and Sandro Stringari Superstripes and the excitation spectrum of a spin-orbit-coupled BEC

Many-body ground state

Three quantum phases γ = (g − g↑↓) / (g + g↑↓) > 0 , n(c) = k20/ (2γg)

−2 −1 0 1 20.6

0.8

1

1.2

1.4x 10

13

k1x / π

| ψ↑,

↓ (

x) |2 [

cm−

3 ]

Ω(I-II)cr = 2k2

0

p

2γ/(1 + 2γ) small for 87Rb

Ω(II-III)cr = 2k2

0

To increase the effect of the contrast, choose larger values of γ

Yun Li, Giovanni I. Martone, Lev P. Pitaevskii, and Sandro Stringari Superstripes and the excitation spectrum of a spin-orbit-coupled BEC

Excitation spectrum in phase II

−2 −1.5 −1 −0.5 0 0.51

2

3

ω+ /

k 02

−2 −1.5 −1 −0.5 0 0.50

0.1

0.2

0.3

q / k0

ω− /

k 02

G1/k20 = 0.5, G2/k

20 = 0.12

Ω/k20 = 1.22, 1.33, 1.46

Despite spinor nature, occurrenceof Raman coupling gives rise to asingle gapless branch

Emergence of a roton minimumat finite q: a tendency of thesystem towards crystallization

Martone, LY, Pitaevskii and Stringari, PRA 86, 063621(2012)

Yun Li, Giovanni I. Martone, Lev P. Pitaevskii, and Sandro Stringari Superstripes and the excitation spectrum of a spin-orbit-coupled BEC

Excitation spectrum in phase II

−2 −1 0 1 20

0.2

0.4

E(p

)

−2 −1.5 −1 −0.5 0 0.50

0.1

0.2

0.3

q / k0

ω− /

k 02

G1/k20 = 0.5, G2/k

20 = 0.12

Ω/k20 = 1.22, 1.33, 1.46

Despite spinor nature, occurrenceof Raman coupling gives rise to asingle gapless branch

Emergence of a roton minimumat finite q: a tendency of thesystem towards crystallization

Martone, LY, Pitaevskii and Stringari, PRA 86, 063621(2012)

Yun Li, Giovanni I. Martone, Lev P. Pitaevskii, and Sandro Stringari Superstripes and the excitation spectrum of a spin-orbit-coupled BEC

Excitation spectrum in phase II

−2 −1 0 1 20

0.2

0.4

E(p

)

−2 −1.5 −1 −0.5 0 0.50

0.1

0.2

0.3

q / k0

ω− /

k 02

G1/k20 = 0.5, G2/k

20 = 0.12

Ω/k20 = 1.22, 1.33, 1.46

Despite spinor nature, occurrenceof Raman coupling gives rise to asingle gapless branch

Emergence of a roton minimumat finite q: a tendency of thesystem towards crystallization

Martone, LY, Pitaevskii and Stringari, PRA 86, 063621(2012)

Yun Li, Giovanni I. Martone, Lev P. Pitaevskii, and Sandro Stringari Superstripes and the excitation spectrum of a spin-orbit-coupled BEC

Excitation spectrum in phase I

Equilibrium state

−2 −1 0 1 20.6

0.8

1

1.2

1.4x 10

13

k1x / π

| ψ↑,

↓ (

x) |2 [

cm−

3 ]

G1/k20 = 0.3, G2/k2

0 = 0.08, Ω/k20 = 1

Translational invariance symmetry breaking

U(1) symmetry breaking

ψ0↑

ψ0↓

«

=P

K

a−k1+K

−b−k1+K

«

ei(K−k1)x, K is the reciprocal lattice vector

Yun Li, Giovanni I. Martone, Lev P. Pitaevskii, and Sandro Stringari Superstripes and the excitation spectrum of a spin-orbit-coupled BEC

Excitation spectrum in phase I

Bogoliubov + Bloch theory

ψ↑

ψ↓

«

= e−iµt

»„

ψ0↑

ψ0↓

«

+

u↑(r)u↓(r)

«

e−iωt +

v∗↑(r)v∗↓(r)

«

eiωt

uq ↑, ↓(r) = e−ik1xX

K

Uq ↑, ↓ K eiq·r+iKx

vq ↑, ↓(r) = eik1xX

K

Vq ↑, ↓ K eiq·r−iKx

0 0.5 1 1.5 20

0.4

0.8

1.2

1.6

qx / k

1

ω /

k 02

Emergence of two gapless bands

Vanishing of the frequency atthe edge of the Brillouin zone

Divergent behavior of staticstructure factor in density channel

Yun Li, Giovanni I. Martone, Lev P. Pitaevskii, and Sandro Stringari Superstripes and the excitation spectrum of a spin-orbit-coupled BEC

Excitation spectrum in phase I

Superfluid Supersolid

a)

0 0.2 0.4 0.6 0.8 1

ka/2π

0

5

10

15

20

ω

a)a)a)a) b)

0 0.2 0.4 0.6 0.8

ka/2π

0

5

10

15

20

25

ω

b)b)b)

Soft-coreinteractions

S. Saccani et al., PRL 108, 175301 (2012)

0 0.5 1 1.5 20

0.4

0.8

1.2

1.6

qx / k

1

ω /

k 02

Emergence of two gapless bands

Vanishing of the frequency atthe edge of the Brillouin zone

Divergent behavior of staticstructure factor in density channel

Yun Li, Giovanni I. Martone, Lev P. Pitaevskii, and Sandro Stringari Superstripes and the excitation spectrum of a spin-orbit-coupled BEC

Density structure factor

0

0.5

1

1.5

2 0

0.4

0.8

1.2

1.60

0.5

1

ω / k02

qx / k

1

S(ω

, qx)

0 0.5 1 1.5 20

0.4

0.8

1.2

1.6

qx / k

1

ω /

k 02

0 0.5 1 1.5 20

0.5

1

qx / k

1

S(q

x)

Yun Li, Giovanni I. Martone, Lev P. Pitaevskii, and Sandro Stringari Superstripes and the excitation spectrum of a spin-orbit-coupled BEC

Density structure factor

0

0.5

1

1.5

2 0

0.4

0.8

1.2

1.60

0.5

1

ω / k02

qx / k

1

S(ω

, qx)

0 0.5 1 1.5 20

0.4

0.8

1.2

1.6

qx / k

1

ω /

k 02

0 0.5 1 1.5 20

0.5

1

qx / k

1

S(q

x)Upper branch isa density waveat small qx

Yun Li, Giovanni I. Martone, Lev P. Pitaevskii, and Sandro Stringari Superstripes and the excitation spectrum of a spin-orbit-coupled BEC

Density structure factor

0

0.5

1

1.5

2 0

0.4

0.8

1.2

1.60

0.5

1

ω / k02

qx / k

1

S(ω

, qx)

0 0.5 1 1.5 20

0.4

0.8

1.2

1.6

qx / k

1

ω /

k 02

0 0.5 1 1.5 20

0.5

1

qx / k

1

S(q

x)Upper branch isa density waveat small qx

S(qx) for thelower branchdiverges

Yun Li, Giovanni I. Martone, Lev P. Pitaevskii, and Sandro Stringari Superstripes and the excitation spectrum of a spin-orbit-coupled BEC

Spin structure factor

0

0.5

1

1.5

2 0

0.4

0.8

1.2

1.60

0.5

1

ω / k02

qx / k

1

Sσ(ω

, qx)

0 0.5 1 1.5 20

0.4

0.8

1.2

1.6

qx / k

1

ω /

k 02

0 0.5 1 1.5 20

0.5

1

qx / k

1

Sσ(q

x)

Yun Li, Giovanni I. Martone, Lev P. Pitaevskii, and Sandro Stringari Superstripes and the excitation spectrum of a spin-orbit-coupled BEC

Spin structure factor

0

0.5

1

1.5

2 0

0.4

0.8

1.2

1.60

0.5

1

ω / k02

qx / k

1

Sσ(ω

, qx)

0 0.5 1 1.5 20

0.4

0.8

1.2

1.6

qx / k

1

ω /

k 02

0 0.5 1 1.5 20

0.5

1

qx / k

1

Sσ(q

x)Lower branch isa spin waveat small qx

Yun Li, Giovanni I. Martone, Lev P. Pitaevskii, and Sandro Stringari Superstripes and the excitation spectrum of a spin-orbit-coupled BEC

Sum rule approach

Define qx-component density operator F and (qx − qB)-componentmomentum density operator G

F =X

j

eiqxxj , 〈 [F , G] 〉 = qxN〈eiqBx〉

G =X

j

h

pxj e−i(qx−qB)xj + e−i(qx−qB)xjpxj

i

/2

p−th moments:

mp(O) =P

ℓ(|〈0|O|ℓ〉|2

+|〈0|O†|ℓ〉|2)ωpℓ0

qx → qB F G

m1 q2x (qx − qB)2

m0 S(qx) |qx − qB |m−1 χ(qx)

Bogoliubov inequality: m−1(F )m1(G) ≥ |〈 [F ,G] 〉|2 χ(qx) ∝ 1/(qx − qB)2

Uncertainty inequality: m0(F )m0(G) ≥ |〈 [F ,G] 〉|2 S(qx) ∝ 1/|qx − qB |

Yun Li, Giovanni I. Martone, Lev P. Pitaevskii, and Sandro Stringari Superstripes and the excitation spectrum of a spin-orbit-coupled BEC

Sound velocity

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1

Ω / k02

c / k

0

Upper branch

Lower branch

I−II phasetransition

c(d, s)⊥

> c(d, s)x , inertia of flow caused by stripes

c(d)⊥

(g + g↑↓)n/2, well reproduced by usual Bogoliubov soundvelocity

Yun Li, Giovanni I. Martone, Lev P. Pitaevskii, and Sandro Stringari Superstripes and the excitation spectrum of a spin-orbit-coupled BEC

Conclusions

Excitation spectrum in the stripephase: double gapless band structure

At small wave vector the lower andupper branches have, respectively, aspin and density nature

Close to the first Brillouin zone thelower branch acquires an importantdensity character, S(qx) diverges

ω /

k 02

qx / k

1

0 0.5 1 1.5 20

0.5

1

1.5

ω /

k 02

qx / k

1

0 0.5 1 1.5 20

0.5

1

1.5

LY, Martone, Pitaevskii, Stringari

arXiv: 1303.6903

Yun Li, Giovanni I. Martone, Lev P. Pitaevskii, and Sandro Stringari Superstripes and the excitation spectrum of a spin-orbit-coupled BEC

Thank you !

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