Abstract

We solve analytically the master equation for a quantized cavity mode, finding the time-dependent reduced density operator. The formalism applies to both amplifiers and attenuator, permitting the results to be converted between them by introducing a temperature that can be negative. We obtain a quasi-probability distribution function similar to that obtained by Glauber [ R. J. Glauber, ed., Quantum Optics ( Academic, New York, 1969)] from our reduced density matrix and equate his parameters with the coefficients appearing in the master equation. We illustrate the formalism by calculating variances, expectation values, and photon statistics for the damped simple harmonic oscillator and the linear amplifier.

© 1993 Optical Society of America

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References

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  1. N. S. Goel and N. Richter-Dyn, Stochastic Models in Biology (Academic, New York, 1974).
  2. A. Ben-Shaul, Y. Haas, K. L. Kompa, and R. D. Levine, Lasers and Chemical Change (Springer-Verlag, Berlin, 1981).
    [Crossref]
  3. M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1977).
  4. V. Weisskopf and E. Wigner, Z. Phys. 63, 54 (1930).
    [Crossref]
  5. P. Meystre and M. Sargent, Elements of Quantum Optics (Springer-Verlag, Berlin, 1990).
    [Crossref]
  6. R. J. Glauber, in Frontiers in Quantum Optics, E. R. Pike and S. Sarkar, eds. (Adam Hilger, Bristol, Mass., 1986).
  7. R. J. Glauber, in Quantum Optics, Proceedings of the International School of Physics, “Enrico Fermi,” Course XLII, R. J. Glauber, ed. (Academic, New York, 1969).
  8. M. Sargent, D. A. Holm, and M. S. Zubairy, Phys. Rev. A 31, 3112 (1985); D. A. Holm, M. Sargent, and S. Stenholm, J. Opt. Soc. Am. B 2, 1456 (1985).
    [Crossref]
  9. B. R. Mollow and R. J. Glauber, Phys. Rev. 160, 1076 (1967).
    [Crossref]
  10. A. B. Balantekin and N. Takigawa, Ann. Phys. (N.Y.) 160, 441 (1985).
    [Crossref]
  11. R. Gilmore and J. M. Yuan, J. Chem. Phys. 86, 130 (1987).
    [Crossref]
  12. F. T. Arecchi, E. Courtens, R. Gilmore, and H. Thomas, Phys. Rev. A 6, 2211 (1972), use matrix representations to discuss Baker–Campbell–Hausdorf relations for the group SU(5).
    [Crossref]
  13. R. Gilmore, Lie Groups, Lie Algebras and Some of Their Applications (Wiley, New York, 1974); K. T. Hecht, The Vector Coherent State Method and Its Application to Problems of Higher Symmetry (Springer-Verlag, Berlin, 1987).
  14. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, San Diego, 1980).
  15. D. R. Truax, Phys. Rev. D 31, 1988 (1985).
    [Crossref]
  16. K. E. Cahill and R. J. Glauber, Phys. Rev. 177, 1857 (1969); Phys. Rev. 177, 1882 (1969).
    [Crossref]
  17. W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1973). See especially Section 6.5.
  18. J. E. Campbell, Proc. London Math. Soc. 28, 381 (1897); H. F. Baker, Proc. London Math. Soc. 34, 347 (1902); F. Hausdorff, Ber. Verh. Saechs. Akad. Wiss. Leipzig Math. Phys. K1. 58, 19 (1906).
  19. T. Schoendorff and H. Risken, Phys. Rev. A 41, 5147 (1990).
    [Crossref] [PubMed]

1990 (1)

T. Schoendorff and H. Risken, Phys. Rev. A 41, 5147 (1990).
[Crossref] [PubMed]

1987 (1)

R. Gilmore and J. M. Yuan, J. Chem. Phys. 86, 130 (1987).
[Crossref]

1985 (3)

D. R. Truax, Phys. Rev. D 31, 1988 (1985).
[Crossref]

M. Sargent, D. A. Holm, and M. S. Zubairy, Phys. Rev. A 31, 3112 (1985); D. A. Holm, M. Sargent, and S. Stenholm, J. Opt. Soc. Am. B 2, 1456 (1985).
[Crossref]

A. B. Balantekin and N. Takigawa, Ann. Phys. (N.Y.) 160, 441 (1985).
[Crossref]

1972 (1)

F. T. Arecchi, E. Courtens, R. Gilmore, and H. Thomas, Phys. Rev. A 6, 2211 (1972), use matrix representations to discuss Baker–Campbell–Hausdorf relations for the group SU(5).
[Crossref]

1969 (1)

K. E. Cahill and R. J. Glauber, Phys. Rev. 177, 1857 (1969); Phys. Rev. 177, 1882 (1969).
[Crossref]

1967 (1)

B. R. Mollow and R. J. Glauber, Phys. Rev. 160, 1076 (1967).
[Crossref]

1930 (1)

V. Weisskopf and E. Wigner, Z. Phys. 63, 54 (1930).
[Crossref]

1897 (1)

J. E. Campbell, Proc. London Math. Soc. 28, 381 (1897); H. F. Baker, Proc. London Math. Soc. 34, 347 (1902); F. Hausdorff, Ber. Verh. Saechs. Akad. Wiss. Leipzig Math. Phys. K1. 58, 19 (1906).

Arecchi, F. T.

F. T. Arecchi, E. Courtens, R. Gilmore, and H. Thomas, Phys. Rev. A 6, 2211 (1972), use matrix representations to discuss Baker–Campbell–Hausdorf relations for the group SU(5).
[Crossref]

Balantekin, A. B.

A. B. Balantekin and N. Takigawa, Ann. Phys. (N.Y.) 160, 441 (1985).
[Crossref]

Ben-Shaul, A.

A. Ben-Shaul, Y. Haas, K. L. Kompa, and R. D. Levine, Lasers and Chemical Change (Springer-Verlag, Berlin, 1981).
[Crossref]

Cahill, K. E.

K. E. Cahill and R. J. Glauber, Phys. Rev. 177, 1857 (1969); Phys. Rev. 177, 1882 (1969).
[Crossref]

Campbell, J. E.

J. E. Campbell, Proc. London Math. Soc. 28, 381 (1897); H. F. Baker, Proc. London Math. Soc. 34, 347 (1902); F. Hausdorff, Ber. Verh. Saechs. Akad. Wiss. Leipzig Math. Phys. K1. 58, 19 (1906).

Courtens, E.

F. T. Arecchi, E. Courtens, R. Gilmore, and H. Thomas, Phys. Rev. A 6, 2211 (1972), use matrix representations to discuss Baker–Campbell–Hausdorf relations for the group SU(5).
[Crossref]

Gilmore, R.

R. Gilmore and J. M. Yuan, J. Chem. Phys. 86, 130 (1987).
[Crossref]

F. T. Arecchi, E. Courtens, R. Gilmore, and H. Thomas, Phys. Rev. A 6, 2211 (1972), use matrix representations to discuss Baker–Campbell–Hausdorf relations for the group SU(5).
[Crossref]

R. Gilmore, Lie Groups, Lie Algebras and Some of Their Applications (Wiley, New York, 1974); K. T. Hecht, The Vector Coherent State Method and Its Application to Problems of Higher Symmetry (Springer-Verlag, Berlin, 1987).

Glauber, R. J.

K. E. Cahill and R. J. Glauber, Phys. Rev. 177, 1857 (1969); Phys. Rev. 177, 1882 (1969).
[Crossref]

B. R. Mollow and R. J. Glauber, Phys. Rev. 160, 1076 (1967).
[Crossref]

R. J. Glauber, in Frontiers in Quantum Optics, E. R. Pike and S. Sarkar, eds. (Adam Hilger, Bristol, Mass., 1986).

R. J. Glauber, in Quantum Optics, Proceedings of the International School of Physics, “Enrico Fermi,” Course XLII, R. J. Glauber, ed. (Academic, New York, 1969).

Goel, N. S.

N. S. Goel and N. Richter-Dyn, Stochastic Models in Biology (Academic, New York, 1974).

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, San Diego, 1980).

Haas, Y.

A. Ben-Shaul, Y. Haas, K. L. Kompa, and R. D. Levine, Lasers and Chemical Change (Springer-Verlag, Berlin, 1981).
[Crossref]

Holm, D. A.

M. Sargent, D. A. Holm, and M. S. Zubairy, Phys. Rev. A 31, 3112 (1985); D. A. Holm, M. Sargent, and S. Stenholm, J. Opt. Soc. Am. B 2, 1456 (1985).
[Crossref]

Kompa, K. L.

A. Ben-Shaul, Y. Haas, K. L. Kompa, and R. D. Levine, Lasers and Chemical Change (Springer-Verlag, Berlin, 1981).
[Crossref]

Lamb, W. E.

M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1977).

Levine, R. D.

A. Ben-Shaul, Y. Haas, K. L. Kompa, and R. D. Levine, Lasers and Chemical Change (Springer-Verlag, Berlin, 1981).
[Crossref]

Louisell, W. H.

W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1973). See especially Section 6.5.

Meystre, P.

P. Meystre and M. Sargent, Elements of Quantum Optics (Springer-Verlag, Berlin, 1990).
[Crossref]

Mollow, B. R.

B. R. Mollow and R. J. Glauber, Phys. Rev. 160, 1076 (1967).
[Crossref]

Richter-Dyn, N.

N. S. Goel and N. Richter-Dyn, Stochastic Models in Biology (Academic, New York, 1974).

Risken, H.

T. Schoendorff and H. Risken, Phys. Rev. A 41, 5147 (1990).
[Crossref] [PubMed]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, San Diego, 1980).

Sargent, M.

M. Sargent, D. A. Holm, and M. S. Zubairy, Phys. Rev. A 31, 3112 (1985); D. A. Holm, M. Sargent, and S. Stenholm, J. Opt. Soc. Am. B 2, 1456 (1985).
[Crossref]

P. Meystre and M. Sargent, Elements of Quantum Optics (Springer-Verlag, Berlin, 1990).
[Crossref]

M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1977).

Schoendorff, T.

T. Schoendorff and H. Risken, Phys. Rev. A 41, 5147 (1990).
[Crossref] [PubMed]

Scully, M. O.

M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1977).

Takigawa, N.

A. B. Balantekin and N. Takigawa, Ann. Phys. (N.Y.) 160, 441 (1985).
[Crossref]

Thomas, H.

F. T. Arecchi, E. Courtens, R. Gilmore, and H. Thomas, Phys. Rev. A 6, 2211 (1972), use matrix representations to discuss Baker–Campbell–Hausdorf relations for the group SU(5).
[Crossref]

Truax, D. R.

D. R. Truax, Phys. Rev. D 31, 1988 (1985).
[Crossref]

Weisskopf, V.

V. Weisskopf and E. Wigner, Z. Phys. 63, 54 (1930).
[Crossref]

Wigner, E.

V. Weisskopf and E. Wigner, Z. Phys. 63, 54 (1930).
[Crossref]

Yuan, J. M.

R. Gilmore and J. M. Yuan, J. Chem. Phys. 86, 130 (1987).
[Crossref]

Zubairy, M. S.

M. Sargent, D. A. Holm, and M. S. Zubairy, Phys. Rev. A 31, 3112 (1985); D. A. Holm, M. Sargent, and S. Stenholm, J. Opt. Soc. Am. B 2, 1456 (1985).
[Crossref]

Ann. Phys. (N.Y.) (1)

A. B. Balantekin and N. Takigawa, Ann. Phys. (N.Y.) 160, 441 (1985).
[Crossref]

J. Chem. Phys. (1)

R. Gilmore and J. M. Yuan, J. Chem. Phys. 86, 130 (1987).
[Crossref]

Phys. Rev. (2)

K. E. Cahill and R. J. Glauber, Phys. Rev. 177, 1857 (1969); Phys. Rev. 177, 1882 (1969).
[Crossref]

B. R. Mollow and R. J. Glauber, Phys. Rev. 160, 1076 (1967).
[Crossref]

Phys. Rev. A (3)

M. Sargent, D. A. Holm, and M. S. Zubairy, Phys. Rev. A 31, 3112 (1985); D. A. Holm, M. Sargent, and S. Stenholm, J. Opt. Soc. Am. B 2, 1456 (1985).
[Crossref]

F. T. Arecchi, E. Courtens, R. Gilmore, and H. Thomas, Phys. Rev. A 6, 2211 (1972), use matrix representations to discuss Baker–Campbell–Hausdorf relations for the group SU(5).
[Crossref]

T. Schoendorff and H. Risken, Phys. Rev. A 41, 5147 (1990).
[Crossref] [PubMed]

Phys. Rev. D (1)

D. R. Truax, Phys. Rev. D 31, 1988 (1985).
[Crossref]

Proc. London Math. Soc. (1)

J. E. Campbell, Proc. London Math. Soc. 28, 381 (1897); H. F. Baker, Proc. London Math. Soc. 34, 347 (1902); F. Hausdorff, Ber. Verh. Saechs. Akad. Wiss. Leipzig Math. Phys. K1. 58, 19 (1906).

Z. Phys. (1)

V. Weisskopf and E. Wigner, Z. Phys. 63, 54 (1930).
[Crossref]

Other (9)

P. Meystre and M. Sargent, Elements of Quantum Optics (Springer-Verlag, Berlin, 1990).
[Crossref]

R. J. Glauber, in Frontiers in Quantum Optics, E. R. Pike and S. Sarkar, eds. (Adam Hilger, Bristol, Mass., 1986).

R. J. Glauber, in Quantum Optics, Proceedings of the International School of Physics, “Enrico Fermi,” Course XLII, R. J. Glauber, ed. (Academic, New York, 1969).

N. S. Goel and N. Richter-Dyn, Stochastic Models in Biology (Academic, New York, 1974).

A. Ben-Shaul, Y. Haas, K. L. Kompa, and R. D. Levine, Lasers and Chemical Change (Springer-Verlag, Berlin, 1981).
[Crossref]

M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1977).

W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1973). See especially Section 6.5.

R. Gilmore, Lie Groups, Lie Algebras and Some of Their Applications (Wiley, New York, 1974); K. T. Hecht, The Vector Coherent State Method and Its Application to Problems of Higher Symmetry (Springer-Verlag, Berlin, 1987).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, San Diego, 1980).

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Equations (64)

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ρ ˙ ( t ) = A 1 [ ρ ( t ) a a a ρ ( t ) a ] B 1 [ a a ρ ( t ) a ρ ( t ) ] + adj ,
A = A 1 + A 1 * , B = B 1 + B 1 * , R = A 1 + B 1 * ,
ρ ˙ ( t ) = R ρ ( t ) a a A a ρ ( t ) a R * a a ρ ( t ) + B a ρ ( t ) a A ρ ( t ) .
B 1 A 1 κ = κ + i κ ,
A = Γ n ¯ , B = Γ ( n ¯ + 1 ) .
n ¯ = 1 exp ( ћ Ω / k B T ) 1 ,
n ¯ = 1 exp ( ћ Ω / k B T ) 1 = 1 B / A 1 , Γ = κ .
A 1 = g 2 D 1 1 + I 2 L 2 [ I 2 L 2 2 I 2 F γ 2 1 2 I 2 L 2 D 1 1 2 D 2 * ( 1 + Γ / i Δ ) 1 + I 2 F ( γ / 2 ) ( D 1 + D 3 * ) ] ,
B 1 = g 2 D 1 1 + I 2 L 2 [ 1 + I 2 L 2 2 I 2 F γ 2 [ ( 1 + 1 2 I 2 L 2 ) D 1 + 1 2 D 2 * ( 1 Γ / i Δ ) 1 + I 2 F ( γ / 2 ) ( D 1 + D 3 * ) ] .
ρ ( t ) = exp [ ϕ ( t ) ] exp [ α ( t ) a ] exp [ χ ( t ) a a ] exp [ α * ( t ) a ]
n | exp [ χ ( t ) a a ] | m = δ n m exp [ χ ( t ) n ] ,
lim x exp [ χ ( t ) a a ] = | 0 0 | .
ρ ( ) = { 1 exp [ χ ( ) ] } exp [ χ ( ) a a ] ,
ρ ( t ) a [ ρ ( t ) ] 1 = [ a α ( t ) ] exp [ χ ( t ) ] ,
ρ ( t ) a [ ρ ( t ) ] 1 = exp [ χ ( t ) ] a + α * ( t ) .
e A B e A = B + [ A , B ] + 1 2 [ A , [ A , B ] ] +
{ A + B e χ + R | α | 2 e χ + [ A ( e χ 1 ) + B ( e χ 1 ) ] a a + α * ( B Re χ ) a } ρ + α [ ( R A e χ ) a ] ρ = [ ϕ ˙ e χ α α ˙ * + ( α ˙ α χ ˙ ) a + e χ α ˙ * a + χ ˙ a a ] ρ .
z ( t ) = exp [ χ ( t ) ] ,
ϕ ˙ ( t ) = A + B [ | α ( t ) | 2 + z ( t ) ] ,
α ˙ * ( t ) = α * ( t ) [ R B z ( t ) ] ,
z ˙ ( t ) = ( A + B ) z ( t ) + B z 2 ( t ) + A .
A B ( | α | 2 + z ) = 0.
α * ( R B z ) = 0.
( A + B ) z B z 2 A = 0.
z 1 , 2 = ( A + B ) ± [ ( A + B ) 2 4 A B ] 1 / 2 2 B = ( A + B ) + ( A B ) 2 B .
α ( ) = 0 , z ( ) = n ¯ / ( n ¯ + 1 ) .
ζ = ( A + B ) / 2 κ .
z ( t ) = A sinh ( κ t ) κ [ cosh ( κ t ) + ζ sinh ( κ t ) ] .
z ( t ) = A κ sinh ( κ t ) cosh ( κ t ) + [ ( A + B ) / 2 κ ] sinh ( κ t ) = 1 exp ( 2 κ t ) B / A exp ( 2 κ t ) .
α * ( t ) α * ( 0 ) = exp [ 0 t ( B z R ) d t ] = exp [ R t + B A κ 0 t sinh ( κ t ) d t cosh ( κ t ) + ζ sinh ( κ t ) ] = exp ( R t ) exp ( B A ζ t κ ( ζ 2 1 ) B A κ 2 ( ζ 2 1 ) × ln { sinh [ κ t + tanh 1 ( 1 / ζ ) ] sinh [ tanh 1 ( 1 / ζ ) ] } ) = exp ( R t ) exp [ 1 2 ( R * + R ) t ] × { sinh [ κ t + tanh 1 ( 1 / ζ ) ] sinh [ tanh 1 ( 1 / ζ ) ] } 1 = exp [ 1 2 ( R R * ) t ] [ ζ sinh ( κ t ) + cosh ( κ t ) ] 1 .
B A ζ κ ( ζ 2 1 ) = 1 2 ( R * + R ) , B A κ 2 ( ζ 2 1 ) = 1.
α * ( t ) = α * ( 0 ) exp [ 1 2 ( R R * ) t ] ζ sinh ( κ t ) + cos ( κ t ) = exp ( i κ t ) cosh ( κ t ) + [ ( A + B ) / 2 κ ] sinh ( κ t ) .
ϕ ˙ = A + B [ | α ( t ) | 2 + z ( t ) ] [ R B z ( t ) ] + 1 2 ( B A ) + B | α ( t ) | 2 + 1 2 ( R R * ) ,
exp [ ϕ ( t ) ] = exp [ ϕ ( 0 ) + κ * t ] exp { 0 t [ R B z ( t ) d t ] } × exp [ B 0 t | α ( t ) | 2 d t ] = exp [ | α ( 0 ) | 2 + κ * t ] α * ( t ) α * ( 0 ) × exp { 0 t | α ( 0 ) | 2 2 B κ d t [ 2 κ cosh ( κ t ) + ( A + B ) sinh ( κ t ) ] 2 } = exp [ | α ( 0 ) | 2 + κ * t ] α * ( t ) α * ( 0 ) × exp [ 2 | α ( 0 ) | 2 B sinh ( κ t ) ( A + B ) sinh ( κ t ) + 2 κ cosh ( κ t ) ] = α * ( t ) α * ( 0 ) exp ( κ * t ) exp { 2 κ | α ( 0 ) | 2 exp [ ( κ t ) ] ( A + B ) sinh ( κ t ) + 2 κ cosh ( κ t ) } = ( 1 z ) exp [ | α ( t ) | 2 z 1 ] .
ρ n m ( t ) n | ρ ( t ) | m = e ϕ n | exp [ α ( t ) a ] exp [ χ ( t ) a a ] exp [ a * ( t ) a ] | m = e ϕ p , q α p α * q q ! p ! n | ( a ) p exp ( χ a a ) a q | m = p = 0 n q = 0 m α p α * q q ! p ! exp [ χ ( m q ) + ϕ ] × δ n p , m q [ m ! n ! ( m q ) ! ( n p ) ! ] 1 / 2 = ( n ! m ! ) 1 / 2 e ϕ α * m n p = 0 n | α | 2 p p ! z n p ( m n p ) = ( n ! m ! ) 1 / 2 exp [ ϕ ( t ) ] [ α * ( t ) ] m n [ z ( t ) ] n L n m n [ | α ( t ) | 2 p z ( t ) ] ,
ρ n n ( t ) = exp [ ϕ ( t ) ] q = 0 n | α ( t ) | 2 q q ! z n q ( t ) ( n q ) .
1 = tr [ ρ ( t ) ] = n ρ n n ( t ) = exp [ ϕ ( t ) ] n = 0 q = 0 n | α ( t ) | 2 q q ! z n q ( t ) ( n n q ) ,
exp [ ϕ ( t ) ] = n = 0 q = 0 n | α ( t ) | 2 q q ! z n q ( t ) ( n q ) .
exp [ ϕ ( t ) ] = q = 0 n = q | α ( t ) | 2 q q ! z n q ( t ) ( n q ) = q = 0 | α ( t ) | 2 q q ! [ 1 z ( t ) ] q + 1 = exp { | α ( t ) | 2 / [ 1 z ( t ) ] } 1 z ( t ) .
Q ( β ) = β | ρ ( t ) | β = d 2 γ P ( γ ) β | γ γ | β = d 2 γ P ( γ ) exp ( | γ β | 2 ) .
Q ( β ) = β | exp [ ϕ ( t ) ] exp [ α ( t ) α ] exp [ χ ( τ ) a a ] exp [ α * ( t ) a ] | β = exp [ ϕ ( t ) ] exp [ α ( t ) β * ] exp { [ z ( t ) 1 ] | β | 2 } exp [ α * ( t ) β ] = [ 1 z ( t ) ] exp { α ( t ) β * + [ z ( t ) 1 ] | β | 2 + α * ( t ) β + | α ( t ) | 2 / [ 1 z ( t ) ] } = [ 1 z ( t ) ] exp { [ 1 z ( t ) ] | β + α ( t ) / [ 1 z ( t ) ] | 2 } .
F [ Q ( β ) ] = F [ P ( β ) ] F [ exp ( | β | 2 ) ] .
F [ Q ( β ) ] = π e ϕ 1 z exp [ ( 5 x i k u ) 2 + ( 5 y i k v ) 2 8 ( 1 z ) ] = π exp [ 8 i ( x k u + y k v ) + k 2 8 ( 1 z ) ] .
F [ exp ( | β | 2 ) ] = π exp ( k 4 / 4 ) .
P ( β ) = F 1 { F [ Q ( β ) ] F [ exp ( | β | 2 ) ] } = F 1 { exp [ 8 i ( x k u + y k v ) + k 2 8 ( z 1 ) ] } = 1 z π z exp { z 1 z [ | α ( t ) | 2 ( z 1 ) 2 + | β | 2 + 2 z 1 ( x u + y ν ) ] } = 1 z π z exp { 1 z z [ | α | 2 ( 1 z ) 2 + | β | 2 + α β * + β α * z 1 ] } = 1 z ( t ) π z ( t ) exp [ 1 z ( t ) z ( t ) | β α ( t ) 1 z ( t ) | 2 ] .
ρ ( t ) = exp [ ϕ ( t ) ] exp [ α ( t ) a ] exp [ χ ( t ) a a ] exp [ α ( t ) a ] = exp [ ϕ ( t ) ] exp [ α ( t ) a ] exp [ χ ( t ) ] × l = 0 ( 1 e χ ) l l ! a l a l exp [ α * ( t ) a ] = exp [ ϕ ( t ) ] exp ( | α | 2 ) exp ( α * a ) × { exp [ χ ( t ) ] l = 0 ( 1 e χ ) l l ! ( a α ) l ( a α * ) l } × exp [ α ( t ) α ] ,
P ( β ) = e ϕ e χ exp ( | α | 2 ) exp ( α * β ) [ l = 0 ( 1 e χ ) l l ! | ( β α ) | 2 l ] × exp [ α ( t ) β * ] = e ϕ z exp ( | α | 2 ) exp ( α * β ) exp [ z 1 z | ( β α ) | 2 ] × exp [ α ( t ) β * ] = e ϕ z exp { | α | 2 z ( z 1 ) [ ( z 1 ) 2 1 ] } exp ( | α | 2 ) × exp { z 1 z [ | β | 2 + β * α z 1 + β α * z 1 + | α | 2 ( 1 z ) 2 | α | ( 1 z ) 2 + | α | 2 ] } = 1 z z exp [ z 1 z | β α ( t ) 1 z | 2 ] .
P ( α , 0 | γ , 0 ) = 1 π n ¯ ( t ) exp [ 1 n ¯ ( t ) | γ α ¯ ( t ) | ] ,
α ¯ ( t ) = α exp ( κ t ) ,
n ¯ ( t ) = n ω [ 1 exp ( 2 κ t ) ]
P ( β ) 1 π n ¯ exp ( | β | 2 n ¯ ) .
a ρ = e ϕ ( e ϕ ρ ) α ,
a a ρ = e ϕ ( χ + α α ) e ϕ ρ ,
a ρ = e ϕ ( e χ α * + α ) e ϕ ρ .
O = tr ( O ρ ) = n n | O ρ | n .
a = e ϕ α tr ( e ϕ ρ ) = e ϕ n = 0 q = 0 n α * | α | 2 ( q 1 ) ( q 1 ) ! z n q ( n q ) = e ϕ α e ϕ ,
a a = e ϕ { z z + α α } tr ( e ϕ ρ ) = e ϕ n = 0 q = 0 n n | α | 2 q q ! z n q ( n q ) = e ϕ ( z z + α α ) e ϕ ,
a = e ϕ ( z α * + α ) tr ( e ϕ ρ ) = e ϕ n = 0 q = 0 n α | α | 2 ( q 1 ) ( q 1 ) ! z n q ( n q ) = e ϕ ( z α * + α ) e ϕ .
a = α * ( t ) 1 z ( t ) = a * ,
a a = n = 1 1 z ( t ) [ z ( t ) + | α ( t ) | 2 1 z ( t ) ] .
( Δ n ) 2 = n 2 n 2 = 1 [ 1 z ( t ) ] 2 { z ( t ) + | α ( t ) | 2 [ 1 + z ( t ) ] 1 z ( t ) } ,
I = ( a + a ) 2 = [ α ( t ) + α * ( t ) ] 2 [ 1 z ( t ) ] 2 + 1 + z ( t ) 1 z ( t ) ,
( Δ X 1 ) 2 = 1 4 ( a + a ) 2 1 4 ( a + a ) 2 = 1 4 1 + z ( t ) 1 z ( t ) .
Q = 1 + z n ( 1 z ) [ 1 + | α ( t ) | 2 ( 1 z ) 2 ] .

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