Abstract

Enlightened by the special transformation in our preceding paper [J. Mod. Opt. 56, 1227 (2009) ], we propose a new complex integration transformation corresponding to two mutually conjugate two-mode entangled states η| and ξ| that is compatible with ηξ phase space quantum mechanics and can be used to obtain the complex fractional Fourier transformation kernel from the chirplet function. This transformation obeys the Parseval theorem and is invertible.

© 2009 Optical Society of America

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  1. V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241-265 (1980).
    [CrossRef]
  2. V. Namias, “Fractionalization of Hankel Transforms,” J. Inst. Math. Appl. 26, 187-197 (1980).
    [CrossRef]
  3. D. Mendlovic and H. M. Ozaktas, “Fractional fourier transforms and their optical implementation:I,” J. Opt. Soc. Am. A 10, 1875-1881 (1993).
    [CrossRef]
  4. H. M. Ozakatas and D. Mendlovic, “Fractional Fourier transforms and their optical implementation:II,” J. Opt. Soc. Am. A 10, 2522-2531 (1993).
    [CrossRef]
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  6. Y. B. Karasik, “Expression of the kernel of a fractional Fourier transform in elementary functions,” Opt. Lett. 19, 769-770 (1993).
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  10. H. Y. Fan, “Fractional Hankel transform studied by charge-amplitude state representations and complex fractional Fourier transformation,” Opt. Lett. 28, 2177-2179 (2003).
    [CrossRef] [PubMed]
  11. H. Y. Fan and H. L. Lu, “Eigenmodes of fractional Hankel transform derived by the entangled-state method,” Opt. Lett. 28, 680-682 (2003).
    [CrossRef] [PubMed]
  12. H. Y. Fan and L. Y. Hu, “Optical transformation from chirplet to fractional Fourier transformation kernel,” J. Mod. Opt. 56, 1227-1229 (2009).
    [CrossRef]
  13. H. Weyl, “Quantenmechanik und Gruppentheorie,” Z. Phys. 46, 1-46 (1927).
    [CrossRef]
  14. A. Einstein, B. Podolsky, and N. Rosen, “Can quantum mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777-780 (1935).
    [CrossRef]
  15. H. Y. Fan and J. R. Klauder, “Eigenvectors of two particles' relative position and total momentum,” Phys. Rev. A 49, 704-707 (1994).
    [CrossRef]
  16. H. Y. Fan and Y. Fan, “EPR entangled states and complex fractional Fourier transformation,” Eur. Phys. J. D 21, 233-238 (2002).
    [CrossRef]
  17. H. Y. Fan, L. Y. Hu, and J. S. Wang, “Eigenfunctions of the complex fractional Fourier transform obtained in the context of quantum optics,” J. Opt. Soc. Am. A 25, 974-978 (2008).
    [CrossRef]
  18. H. Y. Fan, “Time evolution of the Wigner function in the entangled-state representation,” Phys. Rev. A 65, 064102 (2002).
    [CrossRef]
  19. D. Dragoman, “Classical versus complex fractional Fourier transformation,” J. Opt. Soc. Am. A 26, 274-277 (2009).
    [CrossRef]

2009

H. Y. Fan and L. Y. Hu, “Optical transformation from chirplet to fractional Fourier transformation kernel,” J. Mod. Opt. 56, 1227-1229 (2009).
[CrossRef]

D. Dragoman, “Classical versus complex fractional Fourier transformation,” J. Opt. Soc. Am. A 26, 274-277 (2009).
[CrossRef]

2008

2003

2002

H. Y. Fan and Y. Fan, “EPR entangled states and complex fractional Fourier transformation,” Eur. Phys. J. D 21, 233-238 (2002).
[CrossRef]

H. Y. Fan, “Time evolution of the Wigner function in the entangled-state representation,” Phys. Rev. A 65, 064102 (2002).
[CrossRef]

1995

1994

1993

1980

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241-265 (1980).
[CrossRef]

V. Namias, “Fractionalization of Hankel Transforms,” J. Inst. Math. Appl. 26, 187-197 (1980).
[CrossRef]

1935

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777-780 (1935).
[CrossRef]

1927

H. Weyl, “Quantenmechanik und Gruppentheorie,” Z. Phys. 46, 1-46 (1927).
[CrossRef]

Dorsch, R. G.

Dragoman, D.

Einstein, A.

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777-780 (1935).
[CrossRef]

Fan, H. Y.

H. Y. Fan and L. Y. Hu, “Optical transformation from chirplet to fractional Fourier transformation kernel,” J. Mod. Opt. 56, 1227-1229 (2009).
[CrossRef]

H. Y. Fan, L. Y. Hu, and J. S. Wang, “Eigenfunctions of the complex fractional Fourier transform obtained in the context of quantum optics,” J. Opt. Soc. Am. A 25, 974-978 (2008).
[CrossRef]

H. Y. Fan, “Fractional Hankel transform studied by charge-amplitude state representations and complex fractional Fourier transformation,” Opt. Lett. 28, 2177-2179 (2003).
[CrossRef] [PubMed]

H. Y. Fan and H. L. Lu, “Eigenmodes of fractional Hankel transform derived by the entangled-state method,” Opt. Lett. 28, 680-682 (2003).
[CrossRef] [PubMed]

H. Y. Fan, “Time evolution of the Wigner function in the entangled-state representation,” Phys. Rev. A 65, 064102 (2002).
[CrossRef]

H. Y. Fan and Y. Fan, “EPR entangled states and complex fractional Fourier transformation,” Eur. Phys. J. D 21, 233-238 (2002).
[CrossRef]

H. Y. Fan and J. R. Klauder, “Eigenvectors of two particles' relative position and total momentum,” Phys. Rev. A 49, 704-707 (1994).
[CrossRef]

Fan, Y.

H. Y. Fan and Y. Fan, “EPR entangled states and complex fractional Fourier transformation,” Eur. Phys. J. D 21, 233-238 (2002).
[CrossRef]

Hu, L. Y.

H. Y. Fan and L. Y. Hu, “Optical transformation from chirplet to fractional Fourier transformation kernel,” J. Mod. Opt. 56, 1227-1229 (2009).
[CrossRef]

H. Y. Fan, L. Y. Hu, and J. S. Wang, “Eigenfunctions of the complex fractional Fourier transform obtained in the context of quantum optics,” J. Opt. Soc. Am. A 25, 974-978 (2008).
[CrossRef]

Karasik, Y. B.

Klauder, J. R.

H. Y. Fan and J. R. Klauder, “Eigenvectors of two particles' relative position and total momentum,” Phys. Rev. A 49, 704-707 (1994).
[CrossRef]

Lohmann, A. W.

Lu, H. L.

Mendlovic, D.

Namias, V.

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241-265 (1980).
[CrossRef]

V. Namias, “Fractionalization of Hankel Transforms,” J. Inst. Math. Appl. 26, 187-197 (1980).
[CrossRef]

Ozakatas, H. M.

Ozaktas, H. M.

H. M. Ozaktas and D. Mendlovic, “Fourier transforms of fractional orders and their optical interpretation,” Opt. Commun. 101, 163-169 (1993).
[CrossRef]

D. Mendlovic and H. M. Ozaktas, “Fractional fourier transforms and their optical implementation:I,” J. Opt. Soc. Am. A 10, 1875-1881 (1993).
[CrossRef]

Podolsky, B.

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777-780 (1935).
[CrossRef]

Rosen, N.

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777-780 (1935).
[CrossRef]

Wang, J. S.

Weyl, H.

H. Weyl, “Quantenmechanik und Gruppentheorie,” Z. Phys. 46, 1-46 (1927).
[CrossRef]

Appl. Opt.

Eur. Phys. J. D

H. Y. Fan and Y. Fan, “EPR entangled states and complex fractional Fourier transformation,” Eur. Phys. J. D 21, 233-238 (2002).
[CrossRef]

J. Inst. Math. Appl.

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241-265 (1980).
[CrossRef]

V. Namias, “Fractionalization of Hankel Transforms,” J. Inst. Math. Appl. 26, 187-197 (1980).
[CrossRef]

J. Mod. Opt.

H. Y. Fan and L. Y. Hu, “Optical transformation from chirplet to fractional Fourier transformation kernel,” J. Mod. Opt. 56, 1227-1229 (2009).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

H. M. Ozaktas and D. Mendlovic, “Fourier transforms of fractional orders and their optical interpretation,” Opt. Commun. 101, 163-169 (1993).
[CrossRef]

Opt. Lett.

Phys. Rev.

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777-780 (1935).
[CrossRef]

Phys. Rev. A

H. Y. Fan and J. R. Klauder, “Eigenvectors of two particles' relative position and total momentum,” Phys. Rev. A 49, 704-707 (1994).
[CrossRef]

H. Y. Fan, “Time evolution of the Wigner function in the entangled-state representation,” Phys. Rev. A 65, 064102 (2002).
[CrossRef]

Z. Phys.

H. Weyl, “Quantenmechanik und Gruppentheorie,” Z. Phys. 46, 1-46 (1927).
[CrossRef]

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Equations (42)

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d p d q π e 2 i ( p x ) ( q y ) h ( p , q ) f ( x , y ) ,
d x d y π e 2 i ( p x ) ( q y ) f ( x , y ) = h ( p , q ) ,
2 i e i α + 1 d p d q π e 2 i ( p x ) ( q y ) e i x y × exp [ i tan ( π 4 α 2 ) ( p 2 + q 2 ) ] = 1 i sin α e i α exp [ i ( x 2 + y 2 ) 2 tan α i x y sin α ] ,
h ( q , p ) = d u e i p u q + u 2 | H ( Q , P ) | q u 2 = 2 π Tr [ H Δ ( q , p ) ] ,
d p d q π e 2 i ( p x ) ( q y ) h ( p , q ) = 2 π p = x | H ( Q , P ) | y e i x y ,
d x d y π 2 e 2 i ( p x ) ( q y ) p = x | H ( Q , P ) | y e i x y = h ( p , q ) .
| η = exp ( 1 2 | η | 2 + η a η * b + a b ) | 00 , η = η 1 + i η 2 ,
( a b ) | η = η | η , ( b a ) | η = η * | η ;
Q 1 = a + a 2 , Q 2 = b + b 2 ,
P 1 = a a i 2 , P 2 = b b i 2 ,
( Q 1 Q 2 ) | η = 2 η 1 | η , ( P 1 + P 2 ) | η = 2 η 2 | η ,
( Q 1 + Q 2 ) | η = i 2 η 2 | η , ( P 1 P 2 ) | η = i 2 η 1 | η .
d 2 η π | η η | = 1 ,
η | η = π δ ( η η ) δ ( η * η * ) π δ ( 2 ) ( η η ) .
| ξ = exp ( | ξ | 2 2 + ξ a + ξ * b a b ) | 00 , ξ = ξ 1 + i ξ 2 .
( a + b ) | ξ = ξ | ξ , ( a + b ) | ξ = ξ * | ξ ,
( Q 1 + Q 2 ) | ξ = 2 ξ 1 | ξ , ( P 1 P 2 ) | ξ = 2 ξ 2 | ξ .
d 2 ξ π | ξ ξ | = 1 , ξ | ξ = π δ ( 2 ) ( ξ ξ ) .
η | ξ = 1 2 exp [ ( η * ξ ξ * η ) 2 ] ,
d 2 ξ d 2 η π 2 e ( ξ μ ) ( η * ν * ) ( η ν ) ( ξ * μ * ) F ( η , ξ ) D ( ν , μ ) .
d 2 ξ d 2 η π 2 e ( ξ μ ) ( η * ν * ) ( η ν ) ( ξ * μ * ) = d 2 ξ δ ( 2 ) ( ξ μ ) e ν ( ξ * μ * ) ν * ( ξ μ ) = 1 ,
d 2 μ d 2 ν π 2 e ( ξ * μ * ) ( η ν ) ( η * ν * ) ( ξ μ ) D ( ν , μ ) F ( η , ξ ) .
d 2 ξ d 2 η π 2 F ( η , ξ ) d 2 μ d 2 ν π 2 e ( ξ μ ) ( η * ν * ) ( η ν ) ( ξ * μ * ) + ( ξ * μ * ) ( η ν ) ( η * ν * ) ( ξ μ ) = d 2 ξ d 2 η π 2 F ( η , ξ ) e ( ξ η * η ξ * + ξ * η η * ξ ) d 2 μ d 2 ν π 2 e ( η * η * ) μ + ( η η ) μ * e ( ξ * ξ * ) ν + ( ξ ξ ) ν * = d 2 ξ d 2 η F ( η , ξ ) e ( ξ η * η ξ * + ξ * η η * ξ ) δ ( 2 ) ( η η ) δ ( 2 ) ( ξ ξ ) = F ( η , ξ ) .
d 2 ξ d 2 η π 2 | F ( η , ξ ) | 2 = d 2 μ d 2 ν π 2 | D ( ν , μ ) | 2 d 2 μ d 2 ν π 2 exp [ ( μ * ν ν * μ ) + ( μ ν * ν μ * ) ] × d 2 ξ d 2 η π 2 exp [ ( μ * μ * ) η + ( μ μ ) η * + ( ν * ν * ) ξ + ( ν ν ) ξ * ] = d 2 μ d 2 ν π 2 | D ( ν , μ ) | 2 d 2 μ d 2 ν exp [ ( μ * ν ν * μ ) + ( μ ν * ν μ * ) ] δ ( 2 ) ( μ μ ) δ ( 2 ) ( ν ν ) = d 2 μ d 2 ν π 2 | D ( ν , μ ) | 2 .
Δ ( η , ξ ) = d 2 σ π 3 | η σ η + σ | e σ ξ * σ * ξ ;
W ρ ( η , ξ ) = d 2 σ π 3 η + σ | ρ | η σ e σ ξ * σ * ξ .
F ( Q 1 , Q 2 , P 1 , P 2 ) = d 2 η d 2 ξ F ( η , ξ ) Δ ( η , ξ ) ,
F ( η , ξ ) = 4 π 2 Tr [ F ( Q 1 , Q 2 , P 1 , P 2 ) Δ ( η , ξ ) ] = 4 d 2 σ π e σ ξ * σ * ξ η + σ | F ( Q 1 , Q 2 , P 1 , P 2 ) | η σ :
d 2 ξ d 2 η π 2 e ( ξ μ ) ( η * ν * ) ( η ν ) ( ξ * μ * ) F ( η , ξ ) = 4 d 2 ξ d 2 η π 2 e ( ξ μ ) ( η * ν * ) ( η ν ) ( ξ * μ * ) d 2 σ π e σ ξ * σ * ξ η + σ | F ( Q 1 , Q 2 , P 1 , P 2 ) | η σ = 4 d 2 σ d 2 η π e μ ( η * ν * ) + μ * ( η ν ) δ ( 2 ) ( η ν σ ) η + σ | F ( Q 1 , Q 2 , P 1 , P 2 ) | η σ = 4 d 2 σ π e μ * σ μ σ * ν + 2 σ | F ( Q 1 , Q 2 , P 1 , P 2 ) | ν ,
ν + 2 σ | = 2 σ | exp { i 2 [ ν 1 ( P 1 P 2 ) ν 2 ( Q 1 + Q 2 ) ] } ,
ν = ν 1 + i ν 2 .
4 d 2 σ e μ * σ μ σ * ν + 2 σ | = 8 d 2 σ ξ = μ | 2 σ 2 σ | exp { i 2 [ ν 1 ( P 1 P 2 ) ν 2 ( Q 1 + Q 2 ) ] } = 2 π ξ = μ | exp { i 2 [ ν 1 ( P 1 P 2 ) ν 2 ( Q 1 + Q 2 ) ] } = 2 π ξ = μ | e i ( μ 2 ν 1 μ 1 ν 2 ) ,
d 2 ξ d 2 η π 2 e ( ξ μ ) ( η * ν * ) ( η ν ) ( ξ * μ * ) F ( η , ξ ) = 2 π ξ = μ | F ( Q 1 , Q 2 , P 1 , P 2 ) | ν e i ( ν 1 μ 2 ν 2 μ 1 ) .
2 d 2 μ d 2 ν π e ( ξ * μ * ) ( η ν ) ( η * ν * ) ( ξ μ ) ξ = μ | F ( Q 1 , Q 2 , P 1 , P 2 ) | ν e i ( ν 1 μ 2 ν 2 μ 1 ) = F ( η , ξ ) .
4 ( e f + 1 ) 2 exp [ e f 1 e f + 1 ( | η | 2 + | ξ | 2 ) ] exp { f 4 [ ( Q 1 Q 2 ) 2 + ( P 1 + P 2 ) 2 + ( Q 1 + Q 2 ) 2 + ( P 1 P 2 ) 2 4 ] } = exp [ f ( a a + b b ) ] ,
4 ( e f + 1 ) 2 d 2 ξ d 2 η π 2 e ( ξ μ ) ( η * ν * ) ( η ν ) ( ξ * μ * ) exp [ e f 1 e f + 1 ( | η | 2 + | ξ | 2 ) ] = 2 π ξ = μ | exp [ f ( a a + b b ) ] | ν e i ( ν 1 μ 2 ν 2 μ 1 ) .
d 2 ξ d 2 η π 2 e ( ξ μ ) ( η * ν * ) ( η ν ) ( ξ * μ * ) e λ ( | η | 2 + | ξ | 2 ) = 1 1 + λ 2 exp [ λ 1 + λ 2 ( | μ | 2 + | ν | 2 ) + λ 2 1 + λ 2 ( μ ν * μ * ν ) ]
λ = i tan ( π 4 α 2 ) , or f = i ( π 2 α ) , e f = i e i α ,
λ λ 2 + 1 = i 2 tan α , λ 2 λ 2 + 1 = 1 2 1 2 sin α ,
4 ( i e i α + 1 ) 2 d 2 ξ d 2 η π 2 e ( ξ μ ) ( η * ν * ) ( η ν ) ( ξ * μ * ) exp [ i tan ( π 4 α 2 ) ( | η | 2 + | ξ | 2 ) ] = 1 i sin α e i α exp [ i ( | μ | 2 + | ν | 2 ) 2 tan α μ ν * μ * ν 2 sin α + i ( μ 2 ν 1 μ 1 ν 2 ) ] ,
ξ = μ | exp { i ( π 2 α ) ( a a + b b ) } | ν = 1 2 π i sin α e i α exp [ i ( | μ | 2 + | ν | 2 ) 2 tan α μ ν * μ * ν 2 sin α ] ,
ν | exp { i ( π 2 α ) ( a a + b b ) } | ν = d 2 ξ π ν | ξ ξ | exp { i ( π 2 α ) ( a a + b b ) } | ν = 1 2 i sin α e i α d 2 ξ 2 π exp [ ν * ξ ξ * ν 2 + i ( | ν | 2 + | ξ | 2 ) 2 tan α + ξ * ν ν * ξ 2 sin α ] = 1 2 cos α e i α exp ( i | ν | 2 2 tan α ) exp { 1 2 i cot α ( ν * sin α ν * ) ( ν sin α ν ) } ,

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