Transcript Slide 1

Violation of a Bell Inequality Using
Fractional Fourier Transforms
Paulo H S Ribeiro
Instituto de Física - UFRJ
Paraty
August-2007
Quantum Optics Group at IF - UFRJ
Author list
IF/UFRJ
DS Tasca
MP Almeida
SP Walborn
PH Souto Ribeiro
Univ. Brétagne-Sud
P Pellat-Finett
DF/UFMG
CH Monken
Twin Photon Generator
Transverse Momentum Correlations
Motivation
Motivation
Typical Bell-inequality experiment
Bell inequality:
Dichotomic degree of freedom
Dichotomic degree of freedom: 2 orthogonal axis
Exemple: polarization of light
  cos H  sin  V
Probabilities
P  H   cos2 
P(V )  sin 2 
   ,

Bell inequalities
Bell States
CHSH inequalities:
S  E 1 , 1   E 1 ,  2   E  2 , 1   E  2 ,  2   2
Binary base

1,2
1

0 1 12 11 0

2
2


1,2
1

0 1 0 2  1 1 1 2

2
Polarization

1,2
1

H

2
1
V
2
V
1
H
2


1,2
1

H

2
1
H
2
V
1
V
2

Non-separability
Non-separability
Non-separability
EPR correlations between position
and momentum of the photons
S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi
Phys. Rev. Lett. 88, 120401 (2002).
  x2  x1    p2  p1  
2
2
1
4
2
Lu-Ming Duan, G. Giedke, J. I. Cirac, and P. Zoller
Phys. Rev. Lett. 84, 2722 (2002).
  x2  x1     p2  p1   2
2
2

 x2 | x1
  p | 
2
2 p1
2
 0.01
2
Mancini´s inequality
Mancini´s inequality
Duan´s inequality
Dichotomization of transverse
momentum
The fractional Fourier transform
Integral form
iei i ' 2cot 
F   f    ' 
e
sen
e
 i 2cot 
 2i  '/ sen 
e
2
When  = /2  Ordinary Fourier transform
F   f    ' 
2

2
e
 2 i  ' 
f  d
f  d
Optical implementation
Fπ
2
Fπ
Optical implementation of a
Fourier transform. D e ZF are
conjugate planes.
Imaging of the field in plane D
onto plane ZI with unit
magnification.
Optical implementation of a
Fractional Fourier transform of
arbitrary order .
F
z  2 f sin2 ( / 2)
The Fractional Fourier Transform
First works related:
N. Wiener. Hermitian polynomials and fourier analysis. J.
Math. Phys. MIT, 8:70–73, 1929.
E. U. Condon. Immersion of the fourier transform in a
continuous of functional transformations. Proc. Nat. Acad.
Sc. USA, 23:158–164, 1937.
A. L. Patterson. Zeits. Kristal, 112:22–32, 1959.
First application:
Method for solving partial diferential equations 1980;
V. Namias, "The fractional order Fourier transform and its
application to quantum mechanics," J. Inst. Appl. Math. 25,
241–265 (1980).
The Fractional Fourier Transform
Application in physics and engeneering:
R. S. Khare. Fractional fourier analysis of defocused images.
Opt. Comm.,12:386–388, 1974.
A. W. Lohmann, "Image rotation, Wigner rotation and the
fractional Fourier transform," J. Opt. Soc. Am. A 10, 2181–
2186 (1993).
Luís B. Almeida, "The fractional Fourier transform and timefrequency representations," IEEE Trans. Sig. Processing 42
(11), 3084–3091 (1994).
Haldun M. Ozaktas, Zeev Zalevsky and M. Alper Kutay. "The
Fractional Fourier Transform with Applications in Optics and
Signal Processing". John Wiley & Sons (2001). Series in
Pure and Applied Optics.
The Fractional Fourier Transform
Application in physics and engeneering:
Soo-Chang Pei and Jian-Jiun Ding, "Relations between
fractional operations and time-frequency distributions,
and their applications," IEEE Trans. Sig. Processing 49
(8), 1638–1655 (2001).
D. H. Bailey and P. N. Swarztrauber, "The fractional Fourier
transform and applications," SIAM Review 33, 389-404
(1991). (Note that this article refers to the chirp-z
transform variant, not the FRFT.)
The Fractional Fourier Transform
Applications in quantum optics;
Yangjian Cai, Qiang Lin, and Shi-Yao Zhu; Coincidence
fractional Fourier transform with entangled photon pairs
and incoherent light;Appl. Phys. Lett. 86, 021112 (2005)
Fei Wang, Yangjian Cai and Sailing He; Experimental
observation of coincidence fractional Fourier transform with
a partially coherent beam;Opt. Exp., 16, 6999(2006)
Violation of Bell inequalities using the fractional
momentum of the photon.
Fractional momentum analyzer
FFT analyzer
1
2
Fractional momentum analyzer
Rotations in momentum space
v(kx , k y )
F
F
2

2
F  [w( x, y)]
F
w( x,  y)
v(kx , k y )
Angular spectrum
w( x, y)
F
F
2
2
v(kx , k y )
w( x, y)
Transverse amplitude
Clauser-Horne-ShimonyHolt(CHSH) inequality
S  E 1 , 1   E 1 ,  2   E  2 , 1   E  2 ,  2   2
E  ,   
C  ,    C   ,     C   ,    C  ,   
C  ,    C   ,     C   ,    C  ,   
C( , ) is the coincidence counting rate:
Maximal violation:
1  1 

8
,  2  1 

8
, 2   2 

8
Analogy with Polarization
Experimental set-up
DS Tasca et al. arXiv:quant-ph/0605061 - Experiment
.
SP Walborn, et al arXiv:quant-ph/0612141 - Theory
Experimental set-up
FFT orders for the maximal violation









  
  
 3 3 

 5 5 






 ,    2 , 2  ;2 ,2    ,   ; 1 ,  1    4 , 4  ; 2 ,  2    4 , 4 













1

1
Implemented
FFT orders
1  a 

2
;
2  b  c   
4
;
50
3 

;
4 100
5
1   e   f 
4
1   d 
Results
S  E 1, 1   E 1,  2   E 2 , 1   E 2 ,  2   2,44  0,03
E 1 ,  1   0, 44  0, 02;
E 1 ,  2   0, 62  0, 01;
E 2 ,  1   0, 71  0, 01;
E 2 ,  2   0, 67  0, 01
Discussion: positivity of the
Wigner function
Expectation value for the observable F:

dqdp
f  p, q  W  p , q 
2


F 
If f  p, q and W  p, q arepositive  classicaldescription
f  p, q  signal of the fractional momentum
W  p, q     c  dqi dqs f   qi  qs  f   qi  qs  qi qs
Usually f   qi  qs  and f   qi  qs  approximated bygaussians
However f   qi  qs   vP  qi  qs   pump laser´s angular spectrum
and f   qi  qs   sinc  qi  qs   non gaussian  negative Wigner function
Discussion: positivity of the
Wigner function
Conclusion
-We need to show that this negativity is actually responsible for
the violation!
- Anyway a qubit can be buildt with dichotomized transverse
momentum of the photon - A. F. Abouraddy et al., Phys. Rev. A
75, 052114 (2007), arXiv:0708.0653v1 [quant-ph] .
Aditivity of the FFT


,

     ,     
FFT ( )


Advanced wave picture
position-position (correlation)
F F  F2
Max. Count rate:
Min. Count rate:
rc  F2 rs  rs



s
,  c 

 

s
,  c 

s
,  c

s
,  c 
position-momentum (no correlation)
momentum-momentum (anti-correlation)
F F  F
2
2
Min. Count rate:
rc  F rs  rs



     
, 
,

 
2
2c 
s
 2s 2c  

Max. Count rate:
2
     
,
,

 

2
2
2
2
s
c
s
c

 



F F  F3


Equal:
rc  F3 rs  rFs
2
2
     
 s ,   s , 
2c 
2c  

     
 s ,   s , 
2c 
2c  

FFT – Ray optics
Matrix representation of a paraxial ray:

r  
 
Paraxial aproximation:
small angles with the
paraxial direction z.
Paraxial free propagation:
1 z 
Pz  

0
1


Thin lens refraction:
1
Lf   1

 f
0

1

FFT – Ray optics
•Optical system: PROPAGATION + LENS + PROPAGATION
1 z  1
Pz Lf Pz  
   1
0 1 f
01 z 

1   0 1 


z
1 
f
Pz Lf Pz  
 1
 
 f
z2 

2z 
f 
z 

1
f 
FFT – Ray optics
•Fourier Transform (z = f):
Fourier plane
 0
Fπ   1

2
 f

Fourier ray:
 0
rF   1

 f

f
 f 
    




0       

 f 
f

0 

 F 
rF  Fπ r   
 F 
2
Fourier ray scaled coordinate:
F  f 
FFT – Ray optics
Image plane
•Imaging system
 0
Fπ   1

 f

f  0

1
0 

 f

f
   1 0 
0   0 1

Image ray:
  
Fπr  rI  





Aditivity:
Fπ  Fπ Fπ
2
Scaled coordinate of the image ray:
I   
2
FFT – Ray optics
•Arbitrary rotations: Fractional Fourier Transform
 cos( )

F   1
  f ' sin( )

f  f sin( )
'
Fracional focal length:
  cos( )  sin( ) f 
'
f ' sin( ) 

cos( ) 

z  2 f sin2 ( / 2)
F r  r  cos( )r  sin( )rF
FFT – Ray optics
In the paraxial approximation:
sin( ) 
q
q  kx xˆ  k y yˆ
k
ρ  xxˆ  yyˆ
Fractional momentum of order
f
q
ρ  cos( )ρ  sin( ) f
k
'
Mapping the transverse momentum


2
 ρ 
2
f
kx xˆ  k y yˆ 

k
Image
    ρ  ρ