ICO-21 2008

Quantum optical methods in classical optics: Optical realizations of quantum systems

Héctor Moya-Cessa

Instituto Nacional de Astrofísica, Optica y ElectrónicaTonantzintla, Pue

MEXXICO

ICO-21 2008

Talk will cover

• Optical ralization of a quantum invariant

• Optical realization of a quantum beam splitter

• Wigner function to evaluate some divergent series

ICO-21 2008

2

21( ) ,

2

qI q q

2 ( ) 0q t q

2 3( ) 1/t

Time dependent harmonic oscillator (classical)

Ermakov-Lewis Invariant

Ermakov equation Lewis, PRL (1967).

Optical ralization of a quantum invariant

M. Fernández Guasti (Metropolitan University of Mexico)S. Chávez Cerda (INAOE)V. Arrizon (INAOE)

ICO-21 2008

2

2ˆ1ˆ ˆ ˆ( )2

qI p q

Squeezing & displacement

2ln ˆˆ ˆ ˆ ˆ( )22ˆ ˆi qi qp pq

S e D e

Translates into QM as

H. Moya-Cessa and M. Fernández Guasti PHYSICS LETTERS A 311, 1 (2003).

G is an invariant provided its derivative is zero

| ˆ |i Ht

2 22ˆ ˆˆ ( )

2 2

p qH t

ˆ ˆ ˆ| | |SD T

ICO-21 2008

2 2

0 0 2

† †

ˆ ˆ| 1 1ˆ| , ( ) ( )

( ) 2 2

ˆ ˆ ˆ ˆˆ, ,

2 2

p qi H H t n

t t

q ip q ipa a n a a

Time dependence multiplies only one operator

2 22ˆ ˆˆ ( )

2 2

p qH t

ˆ ˆ ˆ| | |SD T

1ˆ( ) ( )

2| ( ) | (0)i n t dt

t e

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1ˆ ˆ| ( ) exp ( ) ( ) | (0)2

t i dt t I T

1ˆ| exp ( ) ( ) | (0)

2i dt t n

†

ˆ| (0) (0) | (0)

ˆ| (0) (0) | (0) | (0)

T

T

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cos[ ( , ) ]A B

A B tA B A B

2ln ˆˆ ˆ ˆ ˆ( )22ˆ ˆi qi qp pq

S e D e

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Praxial 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 2yxp z yp z xE

i g z Ez

,x y

d dp i p i

dx 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

Units such that =1

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2 2 22 2 2 ( )( )

2 2yxp z yp z x

iz

( )i g z dzE e

w w wT S D

2ln( ) 22 , ,

www w

w

i wi wp p w

w wS e D e w x y

22 3

2( ) 1/ , ,w

w w

df z f

dz

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x yT T

1 2( 0) ( ) ( )z G x G y

1 22 2

1 22 2

1 1( ) exp ( ) ( ) exp ( ) ( )

( ) 2 ( ) 2

1 1exp ( ) ( ) exp ( ) ( )

( ) 2 ( ) 2

x y

x y

x x y y

x x y y

dz dzz T i N G x T i N G y

z z

dz dzi I T G x i I T G y

z z

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2

21

( ) ( ) , ( ) ( )2 !

x

n n x n nn

u x H x e N u x nu xn

2

2

21

0

( )exp

2( ) ( )

!

n

nn

x

G x e u xn

Arfken

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Optical realization of a quantum beam splitter

R. Mar Sarao (INAOE)

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Splitting in a 50:50 beam splitter

R.A. Campos, B.E.A. Saleh, and M. C. Teich, Phys. Rev. A 40, 1371 (1989).

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Again paraxial wave equation

Consider the copropagation of two beams, probe and signal, in a Kerr medium. The probe beam produces the index of refraction

If the probe beam has a Gaussian profile, Is astigmatic and slightly tilted,a term xy is produced

S. Chávez-Cerda, J.R.Moya-Cessa, and H. Moya-Cessa, J. of the Opt. Soc. of Am. B 24, 404-407 (2007).

ICO-21 2008

1 1 2 0 0 2( ) ( ) ( ) ( ) ( ) ( )u x u y u x u y u x u y

† †, , , ,2 2q q

q q q q q

q ip q ipa a n a a q x y

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†

0

1( ) ( 1) | ( ) ( ) |n

n

W n D D n

Wigner function as a tool to evaluate divergente series

Roberto de Jesús León (INAOE)E. Martí Panameño (Puebla University)

H. Moya-Cessa and P.L. Knight, Phys. Rev. A 48, 2479-2481 (1993).

Glauber displacement operator† *

( ) a aD e

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Abel: "Divergent series are on the whole devil's work, and it is a shame that one dares to found any proof on them. One can get out of them what one wants if one uses them, and it is they which have made so much unhappiness and so many paradoxes. Can one think of anything more appalling than to say that

where m is a positive number. Here's something to laugh at, friends."

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Conclusions

• It was shown the optical realization of the Lewis-Ermakov invariant and of the “quantum” beam splitter

• The Wigner function was used to evaluate some divergent series.