Abstract

We derive two quantum-mechanical photocount formulas when a light field’s density operator ρ is known; one involves ρ’s coherent state mean value and the other involves ρ’s Wigner function; when this information is known, then using these two formulas to calculate the photocount would be convenient. We employ the technique of integration within an antinormally ordered (or Weyl-ordered) product of operators in our derivation.

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References

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  1. P. L. Kelley and W. H. Kleiner, Phys. Rev. 136, 316 (1964).
    [CrossRef]
  2. M. O. Scully and W. E. Lamb Jr, Phys. Rev. 179, 368 (1969).
    [CrossRef]
  3. B. R. Mollow, Phys. Rev. 168, 1896 (1968).
    [CrossRef]
  4. M. Orszag, Quantum Optics (Springer-Verlag, 2000).
  5. R. Loudon, The Quantum Theory of Light, 2nd ed. (Oxford U. Press, 1983).
  6. H.-Y. Fan, Phys. Lett. A 161, 1 (1991).
    [CrossRef]
  7. H.-Y. Fan, H.-L. Lu, and Y. Fan, Ann. Phys. 321, 480 (2006).
    [CrossRef]
  8. H.-Y. Fan, J. Opt. B: Quantum Semiclassical Opt. 5, R147 (2003).
    [CrossRef]
  9. A. Wünsche, J. Opt. B: Quantum Semiclassical Opt. 1, R11 (1999).
    [CrossRef]
  10. R. J. Glauber, Phys. Rev. 130, 2529 (1963).
    [CrossRef]
  11. R. J. Glauber, Phys. Rev. 131, 2766 (1963).
    [CrossRef]
  12. J. R. Klauder and B. S. Skargerstam, Coherent States (World Scientific, 1985).
  13. W. Magnus, F. Obeerhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd ed. (Springer Verlag, 1966).
  14. E. C. G. Sudarshan, Phys. Rev. Lett. 10, 277 (1963).
    [CrossRef]
  15. W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, 1973).
  16. H.-Y. Fan, Ann. Phys. (2007), doi: 10.1016/j.aop.2007.08.009 [see Eq. (45)].
  17. A. Erdèlyi, Bateman Manuscript Project, Higher Transcendental Functions (McGraw-Hill, 1953).
  18. R. Loudon and O. L. Knight, J. Mod. Opt. 34, 709 (1987).
    [CrossRef]
  19. H.-Y. Fan, J. Phys. A 25, 3443 (1992).
    [CrossRef]
  20. H. Weyl, Z. Phys. 46, 1 (1927).
    [CrossRef]
  21. H. Weyl, The Classical Groups (Princeton U. Press, 1953).
  22. E. P. Wigner, Phys. Rev. 40, 749 (1932).
    [CrossRef]
  23. J. E. Moyal, Proc. Cambridge Philos. Soc. 45, 99 (1949).
    [CrossRef]
  24. R. F. O'Connell and E. Wigner, Phys. Lett. 83, 145 (1981).
    [CrossRef]
  25. M. Hillery, R. Connel, M. Scully, and E. Wigner, Phys. Rep. 106, 121 (1984).
    [CrossRef]
  26. G. S. Agarwal and E. Wolf, Phys. Rev. D 2, 2161 (1970).
    [CrossRef]
  27. V. Buzek, C. H. Keitel, and P. L. Knight, Phys. Rev. A 51, 2575 (1995).
    [CrossRef] [PubMed]
  28. V. V. Dodonov and V. I. Man'ko, Theory of Nonclassical States of Light (Taylor & Francis, 2003).
  29. H.-Y. Fan and H.-L. Lu, Opt. Lett. 31, 2622 (2006).
    [CrossRef] [PubMed]
  30. H.-Y. Fan and H.-L. Lu, Opt. Lett. 28, 680 (2003).
    [CrossRef] [PubMed]

2007 (1)

H.-Y. Fan, Ann. Phys. (2007), doi: 10.1016/j.aop.2007.08.009 [see Eq. (45)].

2006 (2)

H.-Y. Fan, H.-L. Lu, and Y. Fan, Ann. Phys. 321, 480 (2006).
[CrossRef]

H.-Y. Fan and H.-L. Lu, Opt. Lett. 31, 2622 (2006).
[CrossRef] [PubMed]

2003 (2)

H.-Y. Fan and H.-L. Lu, Opt. Lett. 28, 680 (2003).
[CrossRef] [PubMed]

H.-Y. Fan, J. Opt. B: Quantum Semiclassical Opt. 5, R147 (2003).
[CrossRef]

1999 (1)

A. Wünsche, J. Opt. B: Quantum Semiclassical Opt. 1, R11 (1999).
[CrossRef]

1995 (1)

V. Buzek, C. H. Keitel, and P. L. Knight, Phys. Rev. A 51, 2575 (1995).
[CrossRef] [PubMed]

1992 (1)

H.-Y. Fan, J. Phys. A 25, 3443 (1992).
[CrossRef]

1991 (1)

H.-Y. Fan, Phys. Lett. A 161, 1 (1991).
[CrossRef]

1987 (1)

R. Loudon and O. L. Knight, J. Mod. Opt. 34, 709 (1987).
[CrossRef]

1984 (1)

M. Hillery, R. Connel, M. Scully, and E. Wigner, Phys. Rep. 106, 121 (1984).
[CrossRef]

1981 (1)

R. F. O'Connell and E. Wigner, Phys. Lett. 83, 145 (1981).
[CrossRef]

1970 (1)

G. S. Agarwal and E. Wolf, Phys. Rev. D 2, 2161 (1970).
[CrossRef]

1969 (1)

M. O. Scully and W. E. Lamb Jr, Phys. Rev. 179, 368 (1969).
[CrossRef]

1968 (1)

B. R. Mollow, Phys. Rev. 168, 1896 (1968).
[CrossRef]

1964 (1)

P. L. Kelley and W. H. Kleiner, Phys. Rev. 136, 316 (1964).
[CrossRef]

1963 (3)

R. J. Glauber, Phys. Rev. 130, 2529 (1963).
[CrossRef]

R. J. Glauber, Phys. Rev. 131, 2766 (1963).
[CrossRef]

E. C. G. Sudarshan, Phys. Rev. Lett. 10, 277 (1963).
[CrossRef]

1949 (1)

J. E. Moyal, Proc. Cambridge Philos. Soc. 45, 99 (1949).
[CrossRef]

1932 (1)

E. P. Wigner, Phys. Rev. 40, 749 (1932).
[CrossRef]

1927 (1)

H. Weyl, Z. Phys. 46, 1 (1927).
[CrossRef]

Ann. Phys. (2)

H.-Y. Fan, H.-L. Lu, and Y. Fan, Ann. Phys. 321, 480 (2006).
[CrossRef]

H.-Y. Fan, Ann. Phys. (2007), doi: 10.1016/j.aop.2007.08.009 [see Eq. (45)].

J. Mod. Opt. (1)

R. Loudon and O. L. Knight, J. Mod. Opt. 34, 709 (1987).
[CrossRef]

J. Opt. B: Quantum Semiclassical Opt. (2)

H.-Y. Fan, J. Opt. B: Quantum Semiclassical Opt. 5, R147 (2003).
[CrossRef]

A. Wünsche, J. Opt. B: Quantum Semiclassical Opt. 1, R11 (1999).
[CrossRef]

J. Phys. A (1)

H.-Y. Fan, J. Phys. A 25, 3443 (1992).
[CrossRef]

Opt. Lett. (2)

Phys. Lett. (1)

R. F. O'Connell and E. Wigner, Phys. Lett. 83, 145 (1981).
[CrossRef]

Phys. Lett. A (1)

H.-Y. Fan, Phys. Lett. A 161, 1 (1991).
[CrossRef]

Phys. Rep. (1)

M. Hillery, R. Connel, M. Scully, and E. Wigner, Phys. Rep. 106, 121 (1984).
[CrossRef]

Phys. Rev. (6)

E. P. Wigner, Phys. Rev. 40, 749 (1932).
[CrossRef]

R. J. Glauber, Phys. Rev. 130, 2529 (1963).
[CrossRef]

R. J. Glauber, Phys. Rev. 131, 2766 (1963).
[CrossRef]

P. L. Kelley and W. H. Kleiner, Phys. Rev. 136, 316 (1964).
[CrossRef]

M. O. Scully and W. E. Lamb Jr, Phys. Rev. 179, 368 (1969).
[CrossRef]

B. R. Mollow, Phys. Rev. 168, 1896 (1968).
[CrossRef]

Phys. Rev. A (1)

V. Buzek, C. H. Keitel, and P. L. Knight, Phys. Rev. A 51, 2575 (1995).
[CrossRef] [PubMed]

Phys. Rev. D (1)

G. S. Agarwal and E. Wolf, Phys. Rev. D 2, 2161 (1970).
[CrossRef]

Phys. Rev. Lett. (1)

E. C. G. Sudarshan, Phys. Rev. Lett. 10, 277 (1963).
[CrossRef]

Proc. Cambridge Philos. Soc. (1)

J. E. Moyal, Proc. Cambridge Philos. Soc. 45, 99 (1949).
[CrossRef]

Z. Phys. (1)

H. Weyl, Z. Phys. 46, 1 (1927).
[CrossRef]

Other (8)

H. Weyl, The Classical Groups (Princeton U. Press, 1953).

W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, 1973).

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

M. Orszag, Quantum Optics (Springer-Verlag, 2000).

R. Loudon, The Quantum Theory of Light, 2nd ed. (Oxford U. Press, 1983).

J. R. Klauder and B. S. Skargerstam, Coherent States (World Scientific, 1985).

W. Magnus, F. Obeerhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd ed. (Springer Verlag, 1966).

V. V. Dodonov and V. I. Man'ko, Theory of Nonclassical States of Light (Taylor & Francis, 2003).

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Equations (26)

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p ( m , T ) = Tr { ρ : ( ξ a a ) m m ! e ξ a a : } ,
p ( m , T ) = n = m P n ( n m ) ξ m ( 1 ξ ) n m .
A ( a , a ) = d 2 z π z A ( a , a ) z exp [ z 2 + z * a z a + a a ] ,
: ( a a ) m e ξ a a : = d 2 z π z : ( a a ) m e ξ a a : z exp [ z 2 + z * a z a + a a ] = d 2 z π ( ) m z 2 m exp [ ( 1 ξ ) z 2 + z * a z a + a a ] .
: ( a a ) m e ξ a a : = e ξ a a ( ξ 1 ) l = 0 m ( ) l m ! m ! a m l ( a ) m l l ! ( m l ) ! ( m l ) ! ( 1 ξ ) 2 m l + 1 .
L m ( x ) = l = 0 m ( ) l ( m l ) x l l ! ,
: ( a a ) m e ξ a a : = ( ) m m ! ( 1 ξ ) m + 1 e ξ a a ( ξ 1 ) L m ( a a 1 ξ ) ,
A ( a , a ) = d 2 z π A ( z , z * ) z z ,
p ( m , T ) = ( ξ ) m ( 1 ξ ) m + 1 Tr { ρ e ξ a a ( ξ 1 ) L m ( a a 1 ξ ) } = ( ξ ξ 1 ) m Tr { ρ d 2 z ( 1 ξ ) π e ξ z 2 ( 1 ξ ) L m ( z 2 1 ξ ) z z } = ( ξ ξ 1 ) m d 2 z π e ξ z 2 L m ( z 2 ) 1 ξ z ρ 1 ξ z ,
ρ s ( 1 e λ ) D ( α ) S ( r ) e λ a a S 1 ( r ) D 1 ( α ) ,
ρ s D ( α ) e λ a a λ D 1 ( α ) = D ( α ) : exp { a a } : D 1 ( α ) = D ( α ) 0 0 D 1 ( α ) = α α ,
ρ s = 1 σ 1 σ 2 : exp { ( q Q ) 2 2 σ 1 2 ( p P ) 2 2 σ 2 2 } : ,
2 σ 1 2 ( 2 n + 1 ) e 2 r + 1 , 2 σ 2 2 ( 2 n + 1 ) e 2 r + 1 ,
2 n + 1 = 1 + e λ 1 e λ .
L m ( z 2 ) = ( ) m m ! H m , m ( z , z * ) ,
H m , n ( z , z * ) = m + n t m t n exp [ t t + t z + t z * ] t = t = 0 ,
p ( m , T ) = 1 η 1 η 2 m ! ( ξ 1 ξ ) m exp ( ξ q 2 2 η 1 2 ξ p 2 2 η 2 2 ) 2 m t m t m exp { 1 4 ( σ 1 2 η 1 2 σ 2 2 η 2 2 ) ( t 2 + t 2 ) + ( σ 1 2 2 η 1 2 + σ 2 2 2 η 2 2 1 ) t t + 1 ξ 2 η 1 2 η 2 2 [ ( q η 2 2 + i p η 1 2 ) t + ( q η 2 2 i p η 1 2 ) t ] } t = t = 0 .
p ( m , T ) = ( n ξ ) m ( n ξ + 1 ) m + 1 ,
p ( m , T ) = e ξ n ¯ m ! ( ξ n ¯ ) m , ( n ¯ = α a a α = α 2 ) ,
p ( m , T ) = m ! sech r 1 G 2 ( ξ tanh r 1 G 2 ) m k = 0 [ m 2 ] 2 2 k G m 2 k ( m 2 k ) ! k ! k ! ,
p ( m , T ) = ( n ξ ) m ( 1 + n ξ ) m + 1 exp ( ξ α 2 n ξ + 1 ) L m ( α 2 n ( 1 + n ξ ) ) .
A = 2 d 2 z π z A z : : exp [ 2 ( z * a z a + a a ) ] : : .
: ( a a ) m e ξ a a : = : : 2 d 2 z π ( ) m z 2 m exp [ ( 2 ξ ) z 2 + 2 z * a 2 z a + 2 a a ] : : = 2 : : e 2 ξ a a ( ξ 2 ) l = 0 ( ) l m ! m ! ( 2 a ) m l ( 2 a ) m l l ! ( m l ) ! ( m l ) ! ( 2 ξ ) 2 m l + 1 : : = 2 ( ) m m ! ( 2 ξ ) m + 1 : : e 2 ξ a a ( ξ 2 ) L m ( 4 a a 2 ξ ) : : ,
: : e 2 ξ a a ( ξ 2 ) L m ( 4 a a 2 ξ ) : : = 2 d 2 z e 2 ξ z 2 ( ξ 2 ) L m ( 4 z 2 2 ξ ) Δ ( z , z * ) ,
p ( m , T ) = 2 ( ξ ) m ( 2 ξ ) m + 1 d 2 z π e 2 ξ z 2 ( ξ 2 ) L m ( 4 z 2 2 ξ ) W ( z , z * ) ,
d 2 z π exp { ( z * β * ) ( z β ) } H m , n ( z , z * ) = β m β * n ,

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