A omic Da k-B igh Soli ons:
Theo y and Expe imen s
D. J. F an zeskakis*
Depa men o Physics, Uni e si y o A hens, G eece
LENCOS 12, Se ille, July 9-12, 2012
*In collabo a ion wi h:
P. G. Ke ekidis, D. Yan (Amhe s ), R. Ca e e o-González (San Diego),
M. Hoe e (No h Ca olina), P. Engels, J. Chang., C. Hamne (Washing on),
P. Schmelche , S. Middelkamp, J. S ockho e (Hambu g),
J. Cue as, A. Al a ez (Se ille)
V. Achilleos (A hens), V. Ro hos (Thessaloniki)
Ou line
Bοse-Eins ein condensa es (BECs)
- Bina y BECs
- G oss-Pi ae skii mean- ield desc ip ion
Da k-b igh soli ons in bina y BECs
- Single and mul iple da k-b igh soli ons
- SU(2) o a ions: “bea ing” da k-da k soli ons
Da k-b igh soli ons a ini e empe a u es
- Dissipa i e G oss-Pi ae skii equa ions
- Di e en empe a u e-dependen damping egimes
Conclusions and ou look
Bose-Eins ein condensa es (BECs)
BEC: S a e o ma e in which a mac oscopic numbe
o pa icles sha e he same quan um s a e
Theo e ical p edic ion: Bose-Eins ein (1925)
Expe imen al obse a ion:
Co nell-Wieman-Ke e le-Hule (1995)
in ul acold a oms o 87Rb, 23Na and 7Li. Nobel P ize (2001)
Fi s expe imen al obse a ion:
magne ically- apped spin s a es
o 87Rb BEC (JILA g oup, PRL 1997)
Bina y Bose-Eins ein condensa es
2
2
222
2
112
)(
22
2
2
2
1
2
212
2
111
)(
11
2
2
1
2
2
ψψψψψ
ψψψψψ
+++∇−=∂
+++∇−=∂
ggV
m
i
ggV
m
i
ex
ex
Two coupled G oss-Pi ae skii equa ions
( o di e en hype ine s a es o he same a om species):
in a-species a omic collisions in e -species a omic collisions
Typically, e.g., o di e en spin s a es o 87Rb:
221211
ggg
≈≈
Bina y BECs in he mean- ield pic u e
2
2
2
2
12
2
2
1
2
2
2
11
2
1
)(
2
1
1)(
2
1
ψµψψψψ
ψψψψψ
−+++∂−=∂
−+++∂−=∂
zVi
zVi
z
z
1/
2
1
)(
22
<<≡Ω
Ω=
⊥
ωω
z
zzV
Bina y BECs in highly aniso opic (quasi-1D) ha monic aps
Da k-b igh soli ons in homogeneous BECs
kD Dkz zzD
z
i z
)]1(2[
2
1
)(, an)],([
})](exp{i[kzhsec),(
,sin anhcos),(
22
00
2
1
−+−===−=
+=
+=
µθϕζ
θζηψ
ϕζϕψ
“Symbio ic” soli ons
The b igh soli on (suppo ed only by a ac i e in e ac ions)
exis s only due o he in e species in e ac ion wi h he da k soli on
b igh soli on
da k soli on
da k soli on
b igh soli on
z
Hambu g: Phase-imp in ing o a da k soli on in s a e |1, 0>
and illing he densi y dip wi h a oms in s a e |2, 0 >
Washing on: Gene a ion o da k-b igh soli ons by coun e low
o wo componen s in he |1, -1> and |2, -2> s a es
Obse a ions o da k-b igh soli ons
Hambu g expe imen
S. S ellme e al., Na . Phys. 2008
Washing on expe imen s
C. Hamme e al., PRL 2011
S. Middelkamp e al., PLA 2011
Da k-b igh soli ons in he ap
2
2,1
ψ
z
Adiaba ic dynamics o da k-b igh soli ons in he ap - s eps o ollow:
Find an equa ion o he backg ound wa e unc ion
Find a pe u bed NLS sys em o he da k-b igh soli on wa e unc ion
Assume: he soli on’s unc ional o m emains he same
bu he soli on pa ame e s a e unknown unc ions o ime
E olu ion o eno malized Hamil onian e olu ion o soli on pa ame e s
b igh soli on
da k soli on
apping
po en ial
backg ound
densi y
( )
∆
−+=
−−−++∂+∂=
∫
+∞
∞−
µ
ϕ
µ
ψµψψψψ
223
2
2
22
2
2
1
2
2
2
1
sec
2
1
3
4
||)1(2)1|||(|||||
D
N
D
dzE
b
zzDB
Pe u ba ion heo y – Hamil onian app oach
( ) ( )
( ) ( )
uVQuu
z
u
u
i
zdz
dV
VQu
z
i
b
d
2
2
2
2
2
2
2
2
2
2
2
2
1
1
)(||
2
1
12
2
1
)(1||
2
1
υ
µ
υµυ
υ
υυ
µ
υυυ
υυ
−≡=−+−
∂
∂
+
∂
∂
∂
∂
+−≡=−+−
∂
∂
+
∂
∂
App oxima e da k-b igh soli on solu ion o Qd ≠ 0, Qb ≠ 0 as in
he
unpe u bed case bu wi h:
)( an)()(),(),(
0
D z DD
ϕϕϕ
→→→
Coupled GPEs as coupled pe u bed NLSEs
Use o he da k-b igh soli on ene gy o ind he e olu ion
o he unknown ime-dependen soli on pa ame e s D, φ, z0
),()exp()(
1
z iz
υµψ
−Φ=
)(||:ionapp oxima Fe miThomas
2
zV
−=Φ−
µ
BEC wa e unc ion ca ying a da k soli on:
da k soli on
E olu ion o he da k-b igh soli on ene gy:
Oscilla ions o da k-b igh soli ons
dz
Q
QDDNDD
d
dE
b
b
d
db
∂
∂
+
∂
∂
−=++=
∫
+∞
∞−
∗∗
υυ
ϕϕϕ
Re2) an(sec4
2
Equa ion o mo ion o he
da k-b igh soli on cen e :
µ
ωω
b
oscosc
N
z
d
z
≡
+
−
Ω
==+
;
)4/(14
1
2
,0
d
2/1
2
0
2
2
0
2
22
/
Ω
osc
ω
µ
/
b
N
S. Middelkamp e al., Phys. Le . A 2011
Busch-Anglin, PRL 2001
D
N
DDz
b
µ
ϕϕ
2
cos, an
22
0
−==
Dissipa i e dynamics a ini e empe a u es
Coupled dissipa i e G oss-Pi ae skii equa ion (DGPEs)
(Pi ae skii, So . Phys. JETP 1959)
jj
k
kzj j
zVi
ψµψψγ
−++∂−=∂−
∑
=
2
1
2
2
)(
2
1
)(
)(1),(4;T~
µαµαγ
α
>>=<<=
TkTk
BBj
Achilleos, Yan, Ke ekidis, F an zeskakis, NJP 2012
( ) ( )
( ) ( )
∂
∂
+−≡=−+−
∂
∂
+
∂
∂
∂
∂
+
∂
∂
+−≡=−+−
∂
∂
+
∂
∂
u
uVQuu
z
u
u
i
zdz
dV
VQu
z
i
bb
d
d
µγυ
µ
υµυ
υ
µ
γ
υ
υυ
µ
υυυ
υυ
2
2
2
2
2
2
2
2
2
2
2
2
1
1
)(||
2
1
12
2
1
)(1||
2
1
Coupled DGPEs as coupled pe u bed NLSEs
Tempe a u e-induced an idamping o soli ons
22
)2/1( V(z) z
Ω=
Fo adiaba ic pe u ba ion heo y esul s in he
ollowing equa ion o mo ion ( o su icien ly deep/slow soli ons):
0
0
2
00
=+− zzaz
osc
ω
()
oscc c osc
aaaasass
ωω
2,
2
1
0
22
2,1
22
=−±=⇒=+−
+
∈⇒> Rsaa
c 2,1
+
∈=⇒=
Rssaa
c 21
Csaa
c
∈⇒<
2,1
• Supe -c i ical case :
• C i ical case :
• Sub-c i ical case :
Cockbu n, Nis azakis, Ho ikis, Ke ekidis, P oukakis, F an zeskakis, PRL 2010; PRA2011
Achilleos, Yan, Ke ekidis, F an zeskakis, NJP 2012
2/1
2
2
2
2
)4/(4
1
2
,
8
)4/(6
83
2
+
−
Ω
=
+−
+
+
−=
b
b
oscb
b
db
b
b
b
b
d
N
NN
N
NN
a
µ
ωγ
µ
γγ
µ
µ
γ
µ
γµ
Da k and da k-b igh soli on ajec o ies
Supe -c i ical case :Sub-c i ical case :
c
aa
>
c
aa <
Tempe a u e
da k soli on
b igh soli on b igh soli on
da k soli on
da k soli on
da k-b igh soli on
The bi u ca ion diag am is “d i ed” owa ds smalle alues o γ ⇒
da k-b igh soli ons a e mo e obus han da k soli ons
Conclusions
Da k-b igh soli ons in bina y BECs:
- Single- and mul iple-da k-b igh soli ons
- SU(2) o a ions: bea ing and egula da k-da k soli ons
- Dissipa i e dynamics o da k-b igh soli ons a ini e empe a u es
- In all cases connec ion o expe imen s was p o ided
Cu en and u u e wo k
Vec o soli ons in mul i-componen BECs (including spino BECs)
- Sca e ing o ec o soli ons a na ow ba ie s
- Dissipa i e dynamics o ec o soli ons in a ious se ings
- Vec o soli on dynamics in highe -dimensional se ings
“Filled” (wi h he b igh soli on) da k soli ons a e mo e obus han
“ba e” da k soli ons in BECs agains :
empe a u e-induced dissipa ion
ans e se ins abili ies (da k-b igh ings)
Some ele an e e ences:
