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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
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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
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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)
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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
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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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ICO-21 2008
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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ICO-21 2008
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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ICO-21 2008
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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ICO-21 2008
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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ICO-21 2008
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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ICO-21 2008
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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ICO-21 2008
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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ICO-21 2008
Optical realization of a quantum beam splitter
R. Mar Sarao (INAOE)
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ICO-21 2008
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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ICO-21 2008
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ICO-21 2008
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).
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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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ICO-21 2008
†
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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ICO-21 2008
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ICO-21 2008
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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.