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Quantum optical methods in classical optics:
Optical realizations of quantum systems
Héctor Moya-Cessa
Instituto Nacional de Astrofísica, Optica y
Electrónica
Tonantzintla, Pue
MEXXICO
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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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Optical ralization of a quantum invariant
Time dependent harmonic oscillator (classical)

M. Fernández Guasti (Metropolitan
University of Mexico)
S. Chávez Cerda (INAOE)
V. Arrizon (INAOE)
q   (t )q  0
Ermakov-Lewis Invariant
2




1  q 
I     (  q q  )2  ,
2   



Ermakov equation
2
Lewis, PRL (1967).

   2 (t )   1/  3
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Translates into QM as
2
2
ˆ
ˆ
p
q
2
ˆ
H
  (t )
2
2
2



ˆ


1
q
2
ˆI  
ˆ
ˆ
   ( p  q  ) 
2   



Squeezing
Sˆ  e
&
ln 
ˆˆ  pq
ˆ ˆ)
i
( qp
2
displacement

Dˆ  e
i
 2
qˆ
2
G is an invariant
provided its
derivative is zero
H. Moya-Cessa and M. Fernández Guasti
PHYSICS LETTERS A 311, 1 (2003).
 |  ˆ
i
 H | 
t
ˆ ˆ |   Tˆ |  
|   SD
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2
2
ˆ
ˆ
p
q
Hˆ 
  2 (t )
2
2
ˆ ˆ |   Tˆ |  
|   SD
 | 
1 pˆ 2  qˆ 2
1
i
 H 0 |  ,
H0  2
  (t ) (nˆ  )
t
 (t ) 2
2
qˆ  ipˆ
qˆ  ipˆ
†
a
, a 
, nˆ  a † a
2
2
|  (t )  e
1
i ( nˆ  )  ( t ) dt
2

|  (0) 
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Time dependence multiplies
only one operator
|  (0)  Tˆ (0) |  (0) 
†
ˆ
|  (0)  T (0) |  (0) |  (0) 


1
|   exp  i  dt (t ) (nˆ  )  |  (0) 

2 




1 ˆ
ˆ

|  (t )  exp i  dt (t ) ( I  ) T |  (0) 

2 


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A
B


cos[ ( A, B)t ]
A B A B
Sˆ  e
ln 
ˆˆ  pq
ˆ ˆ)
i
( qp
2
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
Dˆ  e
i
 2
qˆ
2
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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k 2 ( x, y )  k12 ( x, y )  12  ( 12 x 2  12 y 2 ), z  L
k ( x, y )  k ( x, y )    ( x   y ), z  L
2
2
2
2
2
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2
2
2
2
2
2
Units such that
=1
2
2
2
2
2
2




p


(
z
)
y
E   px  ( z ) x 
y
i
  
  g ( z )  E
  
z  
2
2
 


2


z  z0
2
1
 ( z)   2
 2 z  z0
d
d

px  i , p y  i
2
dx
dy

1 z  z0
2
 ( z)   2
  2 z  z0
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
g ( z)  

2
1
2
2
z  z0
z  z0
E e 
 i g ( z ) dz

2
2
2
2
2
2


 
p


(
z
)
y
   px  ( z ) x 
y
i
  
  
  
z  
2
2
 
 
Sw  e
ln  w
i
( wpw  pw w )
2

Dw  e
i
w 2
w
2 w
,
Tw  Sw Dw
d 2 w
2
3

f
(
z
)


1/

w
w,
2
dz
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f   ,
w  x, y
  TxTy 
 ( z  0)  G1 ( x)G2 ( y)




dz
1
dz
1


 ( z )  Tx exp i  2 ( N x  ) G1 ( x)Ty exp i  2 ( N y  ) G2 ( y)
  x ( z)
  y ( z)
2
2








dz
1
dz
1


 exp i  2 ( I x  ) TxG1 ( x)exp i  2 ( I y  ) TyG2 ( y)
  x ( z)
  y ( z)
2
2




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un ( x ) 
1
2n n ! 
x2

2
H n ( x)e , N xun ( x)  nun ( x)
Arfken
 (  x)2 
exp 
 2  n
2  2 

G1 ( x) 
 e  un ( x )

n 0 n !
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Optical realization of a quantum beam splitter
R. Mar Sarao (INAOE)
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R.A. Campos, B.E.A. Saleh, and M. C. Teich, Phys. Rev. A 40, 1371 (1989).
Splitting in a 50:50 beam splitter
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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).
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aq 
q  ipq
2
, a 
†
q
q  ipq
2
, nq  aq†aq , q  x, y
u1 ( x)u1 ( y)  u2 ( x)u0 ( y)  u0 ( x)u2 ( y)
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Wigner function as a tool to evaluate divergente series
W ( ) 
1


n
†
(

1)

n
|
D
( )  D( ) | n 

n 0
H. Moya-Cessa and P.L. Knight,
Phys. Rev. A 48, 2479-2481 (1993).
Glauber displacement operator
D( )  e
 a†  *a
Roberto de Jesús León (INAOE)
E. Martí Panameño (Puebla
University)
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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,
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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.
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