Coexistence of Josephson Oscillations and Self Trapping in Optical Waveguide Arrays Ramaz Khomeriki...

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Coexistence of Josephson Coexistence of Josephson Oscillations and Self Trapping Oscillations and Self Trapping in Optical Waveguide Arrays in Optical Waveguide Arrays Ramaz Khomeriki Dipartimento di Energetica ‘‘S. Stecco’’ and CSDC, Università di Firenze, and INFN, Via S. Marta, 3, 50139 Firenze, ITALY Physics Department, Tbilisi State University, 0128 Tbilisi, GEORGIA In collaboration with Jerome Leon (Montpellier, FRANCE) Stefano Ruffo (Firenze, ITALY) Sandro Wimberger (Pisa, ITALY)

Transcript of Coexistence of Josephson Oscillations and Self Trapping in Optical Waveguide Arrays Ramaz Khomeriki...

Page 1: Coexistence of Josephson Oscillations and Self Trapping in Optical Waveguide Arrays Ramaz Khomeriki Dipartimento di Energetica S. Stecco and CSDC, Università

Coexistence of Josephson Oscillations and Coexistence of Josephson Oscillations and Self Trapping in Optical Waveguide ArraysSelf Trapping in Optical Waveguide Arrays

Ramaz Khomeriki

Dipartimento di Energetica ‘‘S. Stecco’’ and CSDC, Università di Firenze, and INFN, Via S. Marta, 3, 50139 Firenze, ITALY

Physics Department, Tbilisi State University, 0128 Tbilisi, GEORGIA

In collaboration with

Jerome Leon (Montpellier, FRANCE)

Stefano Ruffo (Firenze, ITALY)

Sandro Wimberger (Pisa, ITALY)

Page 2: Coexistence of Josephson Oscillations and Self Trapping in Optical Waveguide Arrays Ramaz Khomeriki Dipartimento di Energetica S. Stecco and CSDC, Università

S.M. Jensen, IEEE J. Quantum Electron. v. 18, 1580, (1982).

Page 3: Coexistence of Josephson Oscillations and Self Trapping in Optical Waveguide Arrays Ramaz Khomeriki Dipartimento di Energetica S. Stecco and CSDC, Università

M. Albiez, R. Gati, J. Folling, S. Hunsmann, M. Cristiani, M. K. Oberthaler, PRL, v. 95, 010402 (2005)

Page 4: Coexistence of Josephson Oscillations and Self Trapping in Optical Waveguide Arrays Ramaz Khomeriki Dipartimento di Energetica S. Stecco and CSDC, Università

02 2

2

212

dz

di

02 1

2

121

dz

di

constP 2

2

2

1

P2

2

2

1

2,1

2,12,1 ie

12

cos12

22 P

H

1-P 0; state ground 1PP If 2c Pa

2P1EEnergy state ground Asymmetric 2a

1EEnergy 0 0, state Symmetric s

A. Smerzi, S. Fantoni, S. Giovanazzi, S. R. Shenoy, PRL, 79, 4950, (1997)

Page 5: Coexistence of Josephson Oscillations and Self Trapping in Optical Waveguide Arrays Ramaz Khomeriki Dipartimento di Energetica S. Stecco and CSDC, Università

Q Q

Q

N

Z

0

-N

Page 6: Coexistence of Josephson Oscillations and Self Trapping in Optical Waveguide Arrays Ramaz Khomeriki Dipartimento di Energetica S. Stecco and CSDC, Università

Bose-Einstein Condensates in Optical Lattice

-5 0 5x [2/k

L]

vcos2(kLx/2)

|(x)|2

V0+v

Page 7: Coexistence of Josephson Oscillations and Self Trapping in Optical Waveguide Arrays Ramaz Khomeriki Dipartimento di Energetica S. Stecco and CSDC, Università

02 0

2

0011000

Kc

n

dz

di

022

11 jjjjjjj K

c

n

dz

di

0 ;0 0;j 11 NN Continuous Approximation - Gross-Pitaevskii equation

0)()()()()()( 2

2

2

jjjjVj

jK

z

ji

0V

1 ;

V 00

lNL

cnn j)(V l2

2L

Page 8: Coexistence of Josephson Oscillations and Self Trapping in Optical Waveguide Arrays Ramaz Khomeriki Dipartimento di Energetica S. Stecco and CSDC, Università

-N

0

N

100

200

300

0

0.2

0.4

zz

Waveguide Number [j]z

0 100 200 3000

5

10

z

|j|2

V0

Page 9: Coexistence of Josephson Oscillations and Self Trapping in Optical Waveguide Arrays Ramaz Khomeriki Dipartimento di Energetica S. Stecco and CSDC, Università

)(j),( :Solutions Stationary jez zi

LxlkkLxAj AA for ),()(cn)( A K

lxLkkLxBj BB -for ),()(cn)( B K

lxlebeaj xx -for )(

VA

kA AA ;2

A ;

2

222

;2

B ;

2

222

BkB BB

conditions continuityfour and b, a, B, A, : Parameters Five

Page 10: Coexistence of Josephson Oscillations and Self Trapping in Optical Waveguide Arrays Ramaz Khomeriki Dipartimento di Energetica S. Stecco and CSDC, Università

)()()()(),( 2211 jzjzjz

-N 0 N-0.4

-0.2

0

0.2

0.4

0.6

Symmetric +

Asymmetric a

Antisymmetric _

j

Waveguide Number [j]

2=0.066

2

)()()( ;

2

)()()( 21

jjj

jjj

Page 11: Coexistence of Josephson Oscillations and Self Trapping in Optical Waveguide Arrays Ramaz Khomeriki Dipartimento di Energetica S. Stecco and CSDC, Università

01

2

121

Drdz

di

02

2

212

Drdz

di

j)(

jV(j)jj

21

2121

d

dQ

r

;j)(j)( and 0j :overlap zeroAlmost 22

2121 ddd

j)(

j)(

j)(

j)(2

2

42

21

41

d

d

d

dD

rrD

rD

42K T

4for 1|(0)| and 1|(0)| If 21

Page 12: Coexistence of Josephson Oscillations and Self Trapping in Optical Waveguide Arrays Ramaz Khomeriki Dipartimento di Energetica S. Stecco and CSDC, Università

0 0.5 1 1.5 2 2.5-0.2

-0.15

-0.1

-0.05

0

0.05

P/Pc

(Ea-E

+)/

(Ea+

E+)

Harmonic double well

Square double well

Page 13: Coexistence of Josephson Oscillations and Self Trapping in Optical Waveguide Arrays Ramaz Khomeriki Dipartimento di Energetica S. Stecco and CSDC, Università

-6-4

-20

24 6

0

0.005

0.01

x [2/kL]

|(x)|2

0 100000

0.25

Time [1/ER

]

V0/E

R

10E/ER

5000

10000Time [1/E

R]