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