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, N. Richter-Dyn, Stochastic Models in Biology (Academic, New York, 1974).
  2. A. Ben-Shaul, Y. Haas, K. L. Kompa, R. D. Levine, Lasers and Chemical Change (Springer-Verlag, Berlin, 1981).
    [CrossRef]
  3. M. Sargent, M. O. Scully, W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1977).
  4. V. Weisskopf, E. Wigner, Z. Phys. 63, 54 (1930).
    [CrossRef]
  5. P. Meystre, M. Sargent, Elements of Quantum Optics (Springer-Verlag, Berlin, 1990).
    [CrossRef]
  6. R. J. Glauber, in Frontiers in Quantum Optics, E. R. Pike, 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, M. S. Zubairy, Phys. Rev. A 31, 3112 (1985); D. A. Holm, M. Sargent, S. Stenholm, J. Opt. Soc. Am. B 2, 1456 (1985).
    [CrossRef]
  9. B. R. Mollow, R. J. Glauber, Phys. Rev. 160, 1076 (1967).
    [CrossRef]
  10. A. B. Balantekin, N. Takigawa, Ann. Phys. (N.Y.) 160, 441 (1985).
    [CrossRef]
  11. R. Gilmore, J. M. Yuan, J. Chem. Phys. 86, 130 (1987).
    [CrossRef]
  12. F. T. Arecchi, E. Courtens, R. Gilmore, 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, 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, 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, H. Risken, Phys. Rev. A 41, 5147 (1990).
    [CrossRef] [PubMed]

1990

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

1987

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

1985

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

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

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

1972

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

1969

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

1967

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

1930

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

1897

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, 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, N. Takigawa, Ann. Phys. (N.Y.) 160, 441 (1985).
[CrossRef]

Ben-Shaul, A.

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

Cahill, K. E.

K. E. Cahill, 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, 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, J. M. Yuan, J. Chem. Phys. 86, 130 (1987).
[CrossRef]

F. T. Arecchi, E. Courtens, R. Gilmore, 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, R. J. Glauber, Phys. Rev. 177, 1857 (1969); Phys. Rev. 177, 1882 (1969).
[CrossRef]

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

R. J. Glauber, in Frontiers in Quantum Optics, E. R. Pike, 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, N. Richter-Dyn, Stochastic Models in Biology (Academic, New York, 1974).

Gradshteyn, I. S.

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

Haas, Y.

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

Holm, D. A.

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

Kompa, K. L.

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

Lamb, W. E.

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

Levine, R. D.

A. Ben-Shaul, Y. Haas, K. L. Kompa, 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, M. Sargent, Elements of Quantum Optics (Springer-Verlag, Berlin, 1990).
[CrossRef]

Mollow, B. R.

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

Richter-Dyn, N.

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

Risken, H.

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

Ryzhik, I. M.

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

Sargent, M.

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

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

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

Schoendorff, T.

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

Scully, M. O.

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

Takigawa, N.

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

Thomas, H.

F. T. Arecchi, E. Courtens, R. Gilmore, 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, E. Wigner, Z. Phys. 63, 54 (1930).
[CrossRef]

Wigner, E.

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

Yuan, J. M.

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

Zubairy, M. S.

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

Ann. Phys. (N.Y.)

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

J. Chem. Phys.

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

Phys. Rev.

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

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

Phys. Rev. A

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

F. T. Arecchi, E. Courtens, R. Gilmore, 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, H. Risken, Phys. Rev. A 41, 5147 (1990).
[CrossRef] [PubMed]

Phys. Rev. D

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

Proc. London Math. Soc.

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.

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

Other

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

R. J. Glauber, in Frontiers in Quantum Optics, E. R. Pike, 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, N. Richter-Dyn, Stochastic Models in Biology (Academic, New York, 1974).

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

M. Sargent, M. O. Scully, 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, 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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