Abstract

We employ the recently established basis (the two-variable Hermite–Gaussian function) of the generalized Bargmann space (BGBS) [Phys. Lett. A 303, 311 (2002) ] to study the generalized form of the fractional Fourier transform (FRFT). By using the technique of integration within an ordered product of operators and the bipartite entangled-state representations, we derive the generalized generating function of the BGBS with which the undecomposable kernel of the two-dimensional FRFT [also named complex fractional Fourier transform (CFRFT)] is obtained. This approach naturally shows that the BGBS is just the eigenfunction of the CFRFT.

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References

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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]
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  6. D. Mendlovic, H. M. Ozaktas, and A. W. Lohmann, “Graded index fibers, Wigner distribution functions and fractional Fourier transform,” Appl. Opt. 33, 6188-6193 (1994).
    [CrossRef] [PubMed]
  7. L. M. Bernardo and O. D. D. Soares, “Fractional Fourier transforms and optical systems,” Opt. Commun. 110, 517-522 (1994).
    [CrossRef]
  8. H. M. Ozaktas and D. Mendlovic, “Fourier transforms of fractional order and their optical implementation,” Opt. Commun. 101, 163-169 (1993).
    [CrossRef]
  9. S. Chountasis, A. Vourdas, and C. Bendjaballah, “Fractional Fourier operators and generalized Wigner functions,” Phys. Rev. A 60, 3467-3473 (1999).
    [CrossRef]
  10. I. S. Gradshteyn and L. M. Ryzhik, Tables of Integrals, Series and Products (Academic, 1980).
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    [CrossRef]
  12. H. Y. Fan and J. R. Klauder, “Eigenvectors of two particles' relative position and total momentum,” Phys. Rev. A 49, 704-707 (1994).
    [CrossRef]
  13. H. Y. Fan and Y. Fan, “Representations of two-mode squeezing transformations,” Phys. Rev. A 54, 958-960 (1996).
    [CrossRef]
  14. H. Y. Fan and X. Ye, “Common eigenstates of two particles' center-of-mass coordinates and mass-weighted relative momentum,” Phys. Rev. A 51, 3343-3346 (1995).
    [CrossRef]
  15. H. Y. Fan, “Operator ordering in quantum optics theory and the development of Dirac's symbolic method,” J. Opt. B: Quantum Semiclassical Opt. 5, R147-R163 (2003).
    [CrossRef]
  16. H. Y. Fan, H. R. Zaidi, and J. R. Klauder, “New approach for calculating the normally ordered form of squeeze operators,” Phys. Rev. D 35, 1831-1834 (1987).
    [CrossRef]
  17. A. Wünsche, “About integration within ordered products in quantum optics,” J. Opt. B: Quantum Semiclassical Opt. 2, R11-R21 (2000).
  18. Bateman Manuscript Project, Higher Transcendental Functions, A.Erdèlyi, ed. (McGraw-Hill, 1953).
  19. A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777-780 (1935).
    [CrossRef]
  20. 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]
  21. R. Loudon and P. L. Knight, “Squeezed light,” J. Mod. Opt. 34, 709-759 (1987).
    [CrossRef]
  22. D. F. Walls, “Squeezed states of light,” Nature 306, 141-146 (1983).
    [CrossRef]
  23. H. P. Yuen, “Two-photon coherent states of the radiation field,” Phys. Rev. A 13, 2226-2243 (1976).
    [CrossRef]
  24. M. C. Teich and B. E. A. Saleh, “Squeezed states of light,” Quantum Opt. 1, 153-191 (1989).
    [CrossRef]
  25. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

2003

H. Y. Fan, “Operator ordering in quantum optics theory and the development of Dirac's symbolic method,” J. Opt. B: Quantum Semiclassical Opt. 5, R147-R163 (2003).
[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]

2002

H. Y. Fan and J. H. Chen, “EPR entangled state and generalized Bargmann transformation,” Phys. Lett. A 303, 311-317 (2002).
[CrossRef]

2000

A. Wünsche, “About integration within ordered products in quantum optics,” J. Opt. B: Quantum Semiclassical Opt. 2, R11-R21 (2000).

1999

S. Chountasis, A. Vourdas, and C. Bendjaballah, “Fractional Fourier operators and generalized Wigner functions,” Phys. Rev. A 60, 3467-3473 (1999).
[CrossRef]

1996

H. Y. Fan and Y. Fan, “Representations of two-mode squeezing transformations,” Phys. Rev. A 54, 958-960 (1996).
[CrossRef]

1995

H. Y. Fan and X. Ye, “Common eigenstates of two particles' center-of-mass coordinates and mass-weighted relative momentum,” Phys. Rev. A 51, 3343-3346 (1995).
[CrossRef]

1994

1993

1989

M. C. Teich and B. E. A. Saleh, “Squeezed states of light,” Quantum Opt. 1, 153-191 (1989).
[CrossRef]

1987

H. Y. Fan, H. R. Zaidi, and J. R. Klauder, “New approach for calculating the normally ordered form of squeeze operators,” Phys. Rev. D 35, 1831-1834 (1987).
[CrossRef]

A. C. McBride and F. H. Kerr, “On Namias's fractional Fourier transforms,” IMA J. Appl. Math. 39, 159-175 (1987).
[CrossRef]

R. Loudon and P. L. Knight, “Squeezed light,” J. Mod. Opt. 34, 709-759 (1987).
[CrossRef]

1983

D. F. Walls, “Squeezed states of light,” Nature 306, 141-146 (1983).
[CrossRef]

1980

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

1976

H. P. Yuen, “Two-photon coherent states of the radiation field,” Phys. Rev. A 13, 2226-2243 (1976).
[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]

Appl. Opt.

IMA J. Appl. Math.

A. C. McBride and F. H. Kerr, “On Namias's fractional Fourier transforms,” IMA J. Appl. Math. 39, 159-175 (1987).
[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]

J. Mod. Opt.

R. Loudon and P. L. Knight, “Squeezed light,” J. Mod. Opt. 34, 709-759 (1987).
[CrossRef]

J. Opt. B: Quantum Semiclassical Opt.

H. Y. Fan, “Operator ordering in quantum optics theory and the development of Dirac's symbolic method,” J. Opt. B: Quantum Semiclassical Opt. 5, R147-R163 (2003).
[CrossRef]

A. Wünsche, “About integration within ordered products in quantum optics,” J. Opt. B: Quantum Semiclassical Opt. 2, R11-R21 (2000).

J. Opt. Soc. Am. A

Nature

D. F. Walls, “Squeezed states of light,” Nature 306, 141-146 (1983).
[CrossRef]

Opt. Commun.

L. M. Bernardo and O. D. D. Soares, “Fractional Fourier transforms and optical systems,” Opt. Commun. 110, 517-522 (1994).
[CrossRef]

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

Opt. Lett.

Phys. Lett. A

H. Y. Fan and J. H. Chen, “EPR entangled state and generalized Bargmann transformation,” Phys. Lett. A 303, 311-317 (2002).
[CrossRef]

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 and Y. Fan, “Representations of two-mode squeezing transformations,” Phys. Rev. A 54, 958-960 (1996).
[CrossRef]

H. Y. Fan and X. Ye, “Common eigenstates of two particles' center-of-mass coordinates and mass-weighted relative momentum,” Phys. Rev. A 51, 3343-3346 (1995).
[CrossRef]

S. Chountasis, A. Vourdas, and C. Bendjaballah, “Fractional Fourier operators and generalized Wigner functions,” Phys. Rev. A 60, 3467-3473 (1999).
[CrossRef]

H. P. Yuen, “Two-photon coherent states of the radiation field,” Phys. Rev. A 13, 2226-2243 (1976).
[CrossRef]

Phys. Rev. D

H. Y. Fan, H. R. Zaidi, and J. R. Klauder, “New approach for calculating the normally ordered form of squeeze operators,” Phys. Rev. D 35, 1831-1834 (1987).
[CrossRef]

Quantum Opt.

M. C. Teich and B. E. A. Saleh, “Squeezed states of light,” Quantum Opt. 1, 153-191 (1989).
[CrossRef]

Other

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Bateman Manuscript Project, Higher Transcendental Functions, A.Erdèlyi, ed. (McGraw-Hill, 1953).

I. S. Gradshteyn and L. M. Ryzhik, Tables of Integrals, Series and Products (Academic, 1980).

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Equations (53)

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F α ψ l ( x ) = e i α l ψ l ( x ) .
g ( x ) = l = 0 C l ψ l ( x ) ,
C l = ψ l ( x ) g ( x ) d x .
F α [ g ( x ) ] = l = 0 e i α l ψ l ( x ) C l
= [ l = 0 e i α l ψ l ( x ) ψ l ( x ) ] g ( x ) d x .
K α ( x , x ) = l = 0 e i α l ψ l ( x ) ψ l ( x ) .
l = 0 t l 2 l l ! H l ( x ) H l ( x ) = 1 1 t 2 exp [ t 2 ( x 2 + x 2 ) 2 t x x t 2 1 ] ,
K α ( x 1 , x 1 ) = l = 0 e i α l 2 l l ! H l ( x 1 ) H l ( x 1 ) e ( x 1 2 + x 1 2 ) 2 = e ( x 1 2 + x 1 2 ) 2 1 e i 2 α exp [ e i 2 α ( x 1 2 + x 1 2 ) 2 e i α x 1 x 1 e i 2 α 1 ] = exp [ i ( π 2 α ) ] 2 π sin α exp { i ( x 1 2 + x 1 2 ) 2 tan α i x 1 x 1 sin α } ,
m , n 1 m ! n ! H m , n ( ξ * , ξ ) e 1 2 ξ 2 ,
H m , n ( λ , λ * ) = l = 0 min ( m , n ) ( 1 ) l m ! n ! l ! ( m l ) ! ( n l ) ! λ m l λ * n l ,
[ H m , n ( λ , λ * ) ] * = H m , n ( λ * , λ ) ,
d 2 ξ π e ξ 2 H m , n ( ξ , ξ * ) [ H m , n ( ξ , ξ * ) ] * = m ! n ! m ! n ! δ m , m δ n , n .
ξ m , n = 1 m ! n ! H m , n ( ξ * , ξ ) e ξ 2 2 ,
ξ = exp [ 1 2 ξ 2 + ξ a + ξ * b a b ] 00 ,
ξ = ξ 1 + i ξ 2 ,
( a + b ) ξ = ξ ξ ,
( a + b ) ξ = ξ * ξ ,
( X 1 + X 2 ) ξ = 2 ξ 1 ξ ,
( P 1 P 2 ) ξ = 2 ξ 2 ξ ,
X 1 = a + a 2 , P 1 = a a 2 i ; X 2 = b + b 2 , P 2 = b b 2 i ;
00 00 e a a b b :
d 2 ξ π ξ ξ = d 2 ξ π : exp { [ ξ * ( a + b ) ] [ ξ ( a + b ) ] } 1 .
m , n = 0 t m t n m ! n ! H m , n ( ξ , ξ * ) = exp ( t t + t ξ + t ξ * ) ,
ξ = 00 m , n = 0 a 1 m a 2 n m ! n ! H m , n ( ξ * , ξ ) e ξ 2 2 ,
H m , n ( ξ , ξ * ) = i m + n e ξ 2 d 2 z π z n z * m exp { z 2 i ξ z i ξ * z * }
H m , n ( a + b , a + b ) = d 2 ξ π H m , n ( a + b , a + b ) ξ ξ
= d 2 ξ π ξ ξ H m , n ( ξ , ξ * )
= i m + n d 2 ξ π d 2 z π z n z * m × : exp { z 2 i ξ z i ξ * z * + ξ ( a + b ) }
+ { ξ * ( a + b ) ( a + b ) ( a + b ) } :
= i m + n : d 2 z π z n z * m δ ( a + b i z ) × δ ( a + b i z * ) exp { z 2 ( a + b ) ( a + b ) } :
= : ( a + b ) m ( a + b ) n : .
( a + b ) m ( a + b ) n = d 2 ξ π ξ n ξ * m ξ ξ = d 2 ξ π ξ n ξ * m : exp { [ ξ * ( a + b ) ] [ ξ ( a + b ) ] } : = : H m , n ( a + b , a + b ) : .
( c + d ) v = v v , ( c + d ) v = v * v ,
d 2 v π v v = 1 .
m , n = 0 H m , n ( a + b , a + b ) H m , n ( c + d , c + d ) t n s m m ! n ! = m , n = 0 : [ ( a + b ) ( c + d ) ] m [ ( a + b ) ( c + d ) ] n t n s m m ! n ! : = : exp { s ( a + b ) ( c + d ) + t ( a + b ) ( c + d ) } : .
1 1 t s exp { 1 1 t s [ t s ( a + b ) ( a + b ) + t ( c + d ) ( a + b ) + s ( a + b ) ( c + d ) t s ( c + d ) ( c + d ) ] } U .
U = d 2 ξ π d 2 v π U ( ξ v ) ( ξ v ) = 1 1 t s d 2 ξ π d 2 v π : exp { 1 1 t s [ ξ 2 + t v * ξ * + s ξ v t s v 2 ] v 2 + ξ ( a + b ) + ξ * ( a + b ) ( a + b ) ( a + b ) + v ( c + d ) + v * ( c + d ) ( c + d ) ( c + d ) } : = d 2 v π : exp { v 2 + v [ c + d + s ( a + b ) ] + v * [ c + d + t ( a + b ) ] t s ( a + b ) × ( a + b ) ( c + d ) ( c + d ) } : = : exp { t ( a + b ) ( c + d ) + s ( a + b ) ( c + d ) } : = Eq. ( 24 ) .
m , n = 0 H m , n ( λ , σ ) e σ λ 2 H m , n ( ρ , κ ) e ρ κ 2 t n s m m ! n ! = e ( σ λ ρ κ ) 2 1 1 t s exp { 1 1 t s ( λ σ + t κ σ + s λ ρ t s κ ρ ) } ,
m , n = 0 H m , n ( ξ , ξ * ) e ξ 2 2 H m , n ( i η * , i η ) e η 2 2 t n s m m ! n ! = e ( ξ 2 η 2 ) 2 1 1 t s exp { ξ 2 + i t η ξ * i s η * ξ t s η 2 1 t s } .
m , n = 0 H m , n ( ξ , ξ * ) e ξ 2 2 H m , n ( i η * , i η ) e η 2 2 ( e i α ) n + m m ! n ! = e ξ 2 2 η 2 2 1 1 e 2 i α exp { ξ 2 i η ξ * e i α + i η * ξ e i α e 2 i α η 2 1 e 2 i α } = e i ( π 2 α ) 2 sin α exp { i ( η 2 + ξ 2 ) 2 tan α + ξ * η ξ η * 2 sin α } .
η = [ 1 2 η 2 + η a η * b + a b ] 00 ,
( a b ) η = η η , ( a b ) η = η * η ,
( X 1 X 2 ) η = 2 η 1 η , ( P 1 + P 2 ) η = 2 η 2 η ,
d 2 η π η η = 1 .
η = 00 m , n = 0 i m + n a 1 m a 2 n m ! n ! H m , n ( i η * , i η ) e η 2 2 ,
η m , n = i m + n m ! n ! H m , n ( i η * , i η ) e η 2 2 .
m , n = 0 m , n = 0 η m , n m , n e i ( π 2 + α ) ( a a + b b ) m , n m , n ξ = η e i ( π 2 + α ) ( a a + b b ) ξ ,
F α ( f ( ξ ) ) = e i ( π 2 α ) 2 sin α d 2 ξ π exp { i ( η 2 + ξ 2 ) 2 tan α + ξ * η ξ η * 2 sin α } f ( ξ ) G ( η ) ,
F α ( f ( ξ ) ) = d 2 ξ π η e i ( π 2 + α ) ( a a + b b ) ξ ξ f = η e i ( π 2 + α ) ( a a + b b ) f = η G ,
G = e i ( π 2 + α ) ( a a + b b ) f .
F α ( ξ m , n ) = F α ( 1 m ! n ! H m , n ( ξ * , ξ ) e ξ 2 2 ) = η e i ( π 2 + α ) ( a a + b b ) m , n = e i ( π 2 + α ) ( m + n ) i m + n m ! n ! H m , n ( i η * , i η ) e η 2 2 = i m n e i α ( m + n ) 1 m ! n ! H m , n ( η * , η ) e η 2 2 ,
S 2 = d 2 ξ μ π ξ μ ξ , μ = e λ ,
F α ( ξ S 2 00 ) = d 2 ξ μ π η e i ( π 2 + α ) ( a a + b b ) ξ μ ξ 00 = e i ( π 2 α ) 2 sin α d 2 ξ μ π exp [ ( i ( η 2 + ξ μ 2 ) 2 tan α + ξ * η ξ η * 2 μ sin α ξ 2 2 ) ] = μ e i ( π 2 α ) i cos α + μ 2 sin α exp [ tan α + μ 2 2 ( i + μ 2 tan α ) η 2 ] .

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