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Ljup čo Hadžievski. VINČA Institute of Nuclear Sciences University of Belgrade. Periodic density patterns in dipolar Bose-Einstein condensates trapped in deep optical lattice. Aleksandra Maluckov, Goran Gligori ć, Boris Malomed, Tilman Pfau. GOAL. - PowerPoint PPT Presentation
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Ljupčo Hadžievski
VINČA Institute of Nuclear SciencesUniversity of Belgrade
Aleksandra Maluckov, Goran Gligorić, Boris Malomed, Tilman Pfau
Periodic density patterns in dipolar Bose-Einstein condensates trapped
in deep optical lattice
GOAL
Search for the stable periodic structures in 1D dipolar Bose-Einstein condensates trapped in
deep optical lattices
• Bose-Einstein condensates (BEC)• Dipolar BEC in optical lattice
– Gross-Pitaevskii equation– Dipolar BEC in a cigar-shaped potential (1D)– Dipolar 1D BEC in a deep optical lattice
• Results– Double periodic patterns– Triple periodic patterns
• Conclusion
OUTLINE
Boze-Ajnštajn kondenzati Bose-Einstein condensation is a pure quantum phenomena consisting of the
macroscopic occupation of a single-particle state by an ensemble of identical bosons in thermal equilibrium at finite temperature
1925. -The occurrence of these phenomena was predicted (Einstein-Bose)
1995. -The first successful experimental creation of BECs in dilute alkali gases
2005. - The BEC of Chromium atoms
2008. - The BEC of polar molecules
Dipolar BEC: Significant magnetic or electrical moment of particles
Gross-Pitaevskii equation
ttgVM
tt
i ext ,,2
, 222
rrr
MNag s
24 Feshbach resonance
00 gas Attractive contact interaction
00 gas Repulsive contact interaction
number of atoms
mass of atom
characteristic rangeof magnetic fields
0
1BB
aa rs
Applied magnetic fieldresonant magnetic
fields-wave scattering length
Nonlinearitymanagement
Dipolar BEC
5
22 3r
rgV ddddrereee 121
3
2cos31r
gV dddd
3D Gross-Pitaevskii equation
tdVttgVM
tt
i ddext ,'',',2
, 2222
rrrrrrr
Dipolar contribution
)('
'
'21
3
22
2
2
zfdzzz
zfzfzV
ztzfi
)('
'
'
1
23121
3
2
2
2
2
2
zfdzzz
zf
zf
zfzV
ztzfi
Dipolar BEC in a cigar-shaped potential (1D)
Gross-Pitaevskii equation with the cubic nonlinearity (GPE)
Nonpolynomial nonlinear Schrödinger equation (NPSE)
11Repulsive contact interaction
Attractive contact interaction g
gdd 2cos31
'3
2'2
11'
2nn
nnnnnnn
n
nn
fffffffC
tfi
Discrete Gross-Pitaevskii (DGP) equation (tight-binding approximation))
Discrete 1D model of dipolar BEC- deep optical lattice -
z
+ - + - + -
0 0
Attractive DD interaction
z+
-
+
-
+
-
2 0
Repulsive DD interaction
10 saAttractive contact interaction
10 sa
Repulsive contact interaction
Local nonlinearityNon-local nonlinearity
tinn euf
Discrete 1D model of dipolar BEC- deep optical lattice -
n
nuP 2
Hamiltonian
n nn
nnddnnnDGPE
nn
ffVfgffCH
'3
22'42
1'2
Norm
Conserved quantities
nnn
nnnnnnn U
nn
UUUUUUCU
'3
2'2
11'
2
Stationary solutions
Results
Uniform
Two-periodic
Three-periodic
optical lattice
Patte
rns
T1
T2 =2T1
T3 =3T1
0 3 6 9
-9
-6
-3
0
stableCW
unstable CW
cr
1
0 3 6 9
-9
-6
-3
0
(a)
f 1,f
2
0 3 6 90
1
2
cr
1
(c)
0 3 6 9
0
1
2
(b)
1
cr
f 1,f2
Two-periodic patterns
Stabilitydiagrams
Bifurcationdiagrams
Analytical solutions+
Linear stability analysisUniform
Two-p
eriod
ic
=-2
=-5
t =
-0.5
5
2
cr
0 3 6 9
-9
-6
-3
0
(a)
1
0 3 6 90
1
2
3(b)
2
1
cr
f 1,f
2=f3
0 3 6 9
0
1
2
1
2
cr
(c)
f 1,f 2
=f3
0 3 6 9
-9
-6
-3
0
stableCW
unstable CW
Stabilitydiagrams
Bifurcationdiagrams
Analyt./num. solutions+
Linear stability analysisUniform
Three
-perio
dic
=-2
=-5
Three-periodic patterns
t=-
1.45
0 3 6 9-6
-4
-2
0
CW
DPP
TPP
(a)g
0.0 0.5 1.0 1.5
-4
-2
0 (b)
CW
DPP
TPP
g
Nd
Energy
Three-periodic structures are energetically favorable
Existence and stability are confirmed with the direct numerical simulations
More details in Phys. Rev. Lett. 108, 140402 (2012)
=-2=-5
CONCLUSION
Stable DPP and TPP patterns exist only in the dipolar BEC with repulsive contact and repulsive DD interaction
Challenges: • Experimental verification? (The range of the BEC parameters are experimentally
achievable)• Stable 2D patterns?
+
-+
-
+
-
+
-+
-
+
-+
-
+
-
+
-
+ -
+ -
+ -
+ -
+ - + -
+ -
+ -
+ -
Isotropic DD interactionAnisotropic DD interaction
','2322
2','
''nm
nm
nnmm
f
2','
','2522
22
''
'2'nm
nm nnmm
mmnn f
42 45 48 510
10
20
30
40
50
60
70
80
tps, RDD, =-5, RC, C=0.8linear stability analysis: stablet=0: =-3.9 stat. tps + (rand+reg.) perturbation
n
t[x-1
]
0
0.1714
0.3429
0.5143
0.6857
0.8571
1.029
1.200
21 24 27 30 33 36 39 42 45 48 51 54 57 600
5
10
15
20
25
n
t
00.12380.24760.37140.49520.61900.74290.86670.99051.1141.2381.3621.4861.6101.7331.8571.9812.1052.2292.3522.4762.600
35 40 451.0
1.5
2.0
0
50
100
(a)
ampl
itude
t
n
40 450.0
0.7
1.4
0
30
60
90
(c)am
plitu
de
t
n
+
-+
-
+
-
+
-+
-
+
-+
-
+
-
+
-
+ -
+ -
+ -
+ -
+ - + -
+ -
+ -
+ -
Isotropic DD interactionIDD
Anisotropic DD interactionADD
','2322
2','
''nm
nm
nnmm
f
2','
','2522
22
''
'2'nm
nm nnmm
mmnn f
Dipolar 2D BEC in a deep optical lattice