Ljup čo Hadžievski
description
Transcript of Ljup čo Hadžievski
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