superstripes and the excitation spectrum of a spin-orbit-coupled...
TRANSCRIPT
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
⊥
˜
+Ω
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
⊥
˜
+Ω
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
⊥
˜
+Ω
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
⊥
˜
+Ω
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
⊥
˜
+Ω
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
⊥
˜
+Ω
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
⊥
˜
+Ω
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 !