Classical -quantum analogies : SU(1,1) and Glauber photonic lattices
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Classical-quantum analogies: SU(1,1) and Glauber photonic lattices
Héctor Moya-Cessa
Instituto Nacional de Astrofísica, Optica y ElectrónicaTonantzintla, Pue.
Mexico
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Bessel states as nonlinear coherent states
Infinite waveguides array: fibers
A. L. Jones, JOSA 55, 261-271 (1965)
Weak coupling, interaction only with nearest neighbor:
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1 1 0jz j j
dEik E E
dz
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Waveguide number
05
1 0
Waveguide number1 0
5
A beam injected into one of the waveguides in the array spreads to the rest of them by wave coupling.
Discrete Diffraction
This phenomenon has been referred to as DISCRETE DIFFRACTION
H.S Eisenberg et al, PRL 81, 3383 (1998) ( ) ( ) 2j
j j zE z i J k z
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†
,
| 1|,
| 1 |,
|
n
n
n m
V n n
V n n
n m
†| ( ) | ,
| |
z
jj
d Ei k V V EdZ
E E j
Schrödinger-like equation
1 1 0jz j j
dEik E E
dz
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†
†
( )
1( )
| ( ) | ,
| | (0)
|
(2 ) |
| | (2 ) |
z
z
z
ik V V z
k iV ziV
n nn z
n
nn z
n
d Ei k V V Edz
E e E
e m
i J k V m
j E j i J k m n
†
,
| 1|,
| 1 |,
|
n
n
n m
V n n
V n n
n m
( ) ( ) 2jj j zE z i J k z
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010 E
ZdEd
i
01 11 nnn EnEn
dZEd
i
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)0(exp)( aaiZz )(ZmEm
kaZiaZimZEm )exp()(exp)2/exp( 2
1],[ aa
1( ) ( )
| | 1n na x n x
a n n n
1( ) 1 ( ),
| 1 | 1n na x n x
a n n n
see Arfken
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)()!(
!)()2/(exp)( 22 ZLsk
kZiZZE sk
ssk
)(!
)!()()2/exp()( 22 ZLk
skZiZZE ssk
ssk
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Optics Communications (2011)
f(n)
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SU(1,1) algebra
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221 ( ) ,
2qI q q
2 ( ) 0q t q
2 3( ) 1/t
Classical time dependent HO
Ermakov-Lewis invariant
Ermakov equation
Lewis, PRL (1967).
Optical realization of a quantum invariant
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2
^ 2ˆ1 ˆ ˆ2
qI p q
Squeezing & Displacement
2ln^ ^ ˆˆˆ ˆ ˆ( ) 22i qi qp pq
S e D e
Translation to quantum
H. Moya-Cessa and M. Fernández Guasti PHYSICS LETTERS A 311, 1 (2003).
2 2 21 ˆ ˆ( )2
H p t q
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| |i Ht
2 2†
0 0 2
^ ^
ˆ ˆ| 1 1| , ( )( ) 2 2
2
p qi H H t a at t
q i pa
Time dependence now as a factor
^ ^ ^| | |S D T
^† 1| ( ) exp ( ) | (0)
2t T i dt t a a
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Paraxial wave equation
Suponemos ahora dos medios GRIN pegados
GRaded INdex referring to an optical material with refractive index in the form of a parabolic curve, decreasing from the center towards the cladding.
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2 2 2 2 2 2 21 1 1 1( , ) ( , ) ( ),k x y k x y x y z L
2 2 2 2 2 2 22 2 2 2( , ) ( , ) ( ),k x y k x y x y z L
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2 2 22 2 2 ( )( ) ( )2 2
yxp z yp z xEi g z E
z
,x yd dp i p idx dy
22 1 0
22 0
22 1 0
22 0
21 022 0
( )
( )
( )
z zz
z z
z zz
z z
z zg z
z z
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2 2 22 2 2 ( )( )2 2
yxp z yp z xi
z
( )i g z dzE e
w w wT S D
2ln( ) 22 , ,
www w
wi wi wp p w
w wS e D e w x y
22 3
2 ( ) 1/xx x
dz
dz
2
2 32 ( ) 1/y
y y
dz
dz
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x yT T
1 2( 0) ( ) ( )z G x G y
1 22 2
1 1( ) exp ( ) ( ) exp ( ) ( )( ) 2 ( ) 2
x y
x x y ydz dzz T i N G x T i N G y
z z
H. MOYA-CESSA, M. Fernández Guasti, V.M. Arrizon and S. Chávez-Cerda, Opt. Lett. 34, No. 9, 1459-1461 (2009), “OPTICAL REALIZATION OF QUANTUM MECHANICAL INVARIANTS.”