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

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

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)

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.”