Abstract

We investigate the representation of mixed states in quantum optics by pure states in an expanded-state space. This thermofield representation is used to describe interactions involving thermal averages and, in particular, interactions involving parametric couplings. We find the thermofield representation of utility in the analysis of non-classical aspects of photon statistics, such as two-mode squeezing of light fluctuations.

© 1985 Optical Society of America

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  1. Y. Takahashi and H. Umezawa, Collect. Phenom. 2, 55 (1975).
  2. We have used the thermal-density matrix ρth= Z−1(β) e−βH.See, for example, the review by D. Ter Haar, Rep. Prog. Phys. 24, 304 (1961).
    [Crossref]
  3. W. H. Louisell demonstrated the use of two-mode expectation values to evaluate thermal averages of single-mode Bose operators in Quantum Statistical Properties of Radiation (Wiley, New York, 1973), p. 161.
  4. R. J. Glauber, Phys. Rev. 131, 2766 (1963).
    [Crossref]
  5. G. J. Milburn, J. Phys. (Paris) A 17, 737 (1984).
  6. H. P. Yuen, Phys. Rev. A 13, 2226 (1976).
    [Crossref]
  7. D. F. Walls, Nature 306, 141 (1983).
    [Crossref]
  8. G. Lachs, Phys. Rev. 138, B1012 (1965).
    [Crossref]
  9. E. T. Jaynes and F. W. Cummings, Proc. IEEE 51, 89 (1963).
    [Crossref]
  10. P. L. Knight and P. M. Radmore, Phys. Rev. A 26, 676 (1982).
    [Crossref]
  11. N. B. Narozhny, J. J. Sanchez-Mondragon, and J. H. Eberly, Phys. Rev. A 23, 236 (1981), and references therein.
    [Crossref]
  12. P. L. Knight and P. M. Radmore, Phys. Lett. 90A, 342 (1982).
  13. See, for example, W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1973), p. 418;H. Haken, Light (North-Holland, Amsterdam, 1981), Vol. 1, p. 285.
  14. W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1973), p. 336.
  15. H. J. Kreuzer and C. G. Kuper, Collect. Phenom. 2, 141 (1976).
  16. H. J. Kreuzer and C. G. Kuper, Collect. Phenom. 2, 131 (1976);J. Phys. G 2, 9 (1976).
  17. W. H. Louisell, A. Yariv, and A. E. Siegman, Phys. Rev. 124, 1646 (1961).
    [Crossref]
  18. B. R. Mollow and R. J. Glauber, Phys. Rev. 160, 1076 (1976).
    [Crossref]
  19. B. R. Mollow and R. J. Glauber, Phys. Rev. 160, 1097 (1976).
    [Crossref]
  20. If the P function is moresingular than a δ function, we say that it ceases to exist. This is the criterion used by Mollow and Glauber.18,19
  21. S. M. Barnett and P. L. Knight, Opt. Acta 31, 435 (1984), and references therein.
    [Crossref]
  22. S. Kumar and C. L. Mehta, Phys. Rev. A 21, 1573 (1980);Phys. Rev. A 24, 1460 (1981).
    [Crossref]
  23. W. Israel, Phys. Lett. 57A, 107 (1976).
  24. S. W. Hawking, Nature 248, 30 (1974);Commun. Math. Phys. 43, 199 (1975);P. C. W. Davies, J. Phys. (Paris) A 8, 609 (1975);Rep. Prog. Phys. 41, 1313 (1978);W. G. Unruh, Phys. Rev. D 14, 870 (1976).
    [Crossref]
  25. L. Parker, Phys. Rev. 183, 1057 (1969).
    [Crossref]

1984 (2)

G. J. Milburn, J. Phys. (Paris) A 17, 737 (1984).

S. M. Barnett and P. L. Knight, Opt. Acta 31, 435 (1984), and references therein.
[Crossref]

1983 (1)

D. F. Walls, Nature 306, 141 (1983).
[Crossref]

1982 (2)

P. L. Knight and P. M. Radmore, Phys. Rev. A 26, 676 (1982).
[Crossref]

P. L. Knight and P. M. Radmore, Phys. Lett. 90A, 342 (1982).

1981 (1)

N. B. Narozhny, J. J. Sanchez-Mondragon, and J. H. Eberly, Phys. Rev. A 23, 236 (1981), and references therein.
[Crossref]

1980 (1)

S. Kumar and C. L. Mehta, Phys. Rev. A 21, 1573 (1980);Phys. Rev. A 24, 1460 (1981).
[Crossref]

1976 (6)

W. Israel, Phys. Lett. 57A, 107 (1976).

H. J. Kreuzer and C. G. Kuper, Collect. Phenom. 2, 141 (1976).

H. J. Kreuzer and C. G. Kuper, Collect. Phenom. 2, 131 (1976);J. Phys. G 2, 9 (1976).

H. P. Yuen, Phys. Rev. A 13, 2226 (1976).
[Crossref]

B. R. Mollow and R. J. Glauber, Phys. Rev. 160, 1076 (1976).
[Crossref]

B. R. Mollow and R. J. Glauber, Phys. Rev. 160, 1097 (1976).
[Crossref]

1975 (1)

Y. Takahashi and H. Umezawa, Collect. Phenom. 2, 55 (1975).

1974 (1)

S. W. Hawking, Nature 248, 30 (1974);Commun. Math. Phys. 43, 199 (1975);P. C. W. Davies, J. Phys. (Paris) A 8, 609 (1975);Rep. Prog. Phys. 41, 1313 (1978);W. G. Unruh, Phys. Rev. D 14, 870 (1976).
[Crossref]

1969 (1)

L. Parker, Phys. Rev. 183, 1057 (1969).
[Crossref]

1965 (1)

G. Lachs, Phys. Rev. 138, B1012 (1965).
[Crossref]

1963 (2)

E. T. Jaynes and F. W. Cummings, Proc. IEEE 51, 89 (1963).
[Crossref]

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

1961 (2)

We have used the thermal-density matrix ρth= Z−1(β) e−βH.See, for example, the review by D. Ter Haar, Rep. Prog. Phys. 24, 304 (1961).
[Crossref]

W. H. Louisell, A. Yariv, and A. E. Siegman, Phys. Rev. 124, 1646 (1961).
[Crossref]

Barnett, S. M.

S. M. Barnett and P. L. Knight, Opt. Acta 31, 435 (1984), and references therein.
[Crossref]

Cummings, F. W.

E. T. Jaynes and F. W. Cummings, Proc. IEEE 51, 89 (1963).
[Crossref]

Eberly, J. H.

N. B. Narozhny, J. J. Sanchez-Mondragon, and J. H. Eberly, Phys. Rev. A 23, 236 (1981), and references therein.
[Crossref]

Glauber, R. J.

B. R. Mollow and R. J. Glauber, Phys. Rev. 160, 1076 (1976).
[Crossref]

B. R. Mollow and R. J. Glauber, Phys. Rev. 160, 1097 (1976).
[Crossref]

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

Hawking, S. W.

S. W. Hawking, Nature 248, 30 (1974);Commun. Math. Phys. 43, 199 (1975);P. C. W. Davies, J. Phys. (Paris) A 8, 609 (1975);Rep. Prog. Phys. 41, 1313 (1978);W. G. Unruh, Phys. Rev. D 14, 870 (1976).
[Crossref]

Israel, W.

W. Israel, Phys. Lett. 57A, 107 (1976).

Jaynes, E. T.

E. T. Jaynes and F. W. Cummings, Proc. IEEE 51, 89 (1963).
[Crossref]

Knight, P. L.

S. M. Barnett and P. L. Knight, Opt. Acta 31, 435 (1984), and references therein.
[Crossref]

P. L. Knight and P. M. Radmore, Phys. Lett. 90A, 342 (1982).

P. L. Knight and P. M. Radmore, Phys. Rev. A 26, 676 (1982).
[Crossref]

Kreuzer, H. J.

H. J. Kreuzer and C. G. Kuper, Collect. Phenom. 2, 141 (1976).

H. J. Kreuzer and C. G. Kuper, Collect. Phenom. 2, 131 (1976);J. Phys. G 2, 9 (1976).

Kumar, S.

S. Kumar and C. L. Mehta, Phys. Rev. A 21, 1573 (1980);Phys. Rev. A 24, 1460 (1981).
[Crossref]

Kuper, C. G.

H. J. Kreuzer and C. G. Kuper, Collect. Phenom. 2, 131 (1976);J. Phys. G 2, 9 (1976).

H. J. Kreuzer and C. G. Kuper, Collect. Phenom. 2, 141 (1976).

Lachs, G.

G. Lachs, Phys. Rev. 138, B1012 (1965).
[Crossref]

Louisell, W. H.

W. H. Louisell, A. Yariv, and A. E. Siegman, Phys. Rev. 124, 1646 (1961).
[Crossref]

W. H. Louisell demonstrated the use of two-mode expectation values to evaluate thermal averages of single-mode Bose operators in Quantum Statistical Properties of Radiation (Wiley, New York, 1973), p. 161.

See, for example, W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1973), p. 418;H. Haken, Light (North-Holland, Amsterdam, 1981), Vol. 1, p. 285.

W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1973), p. 336.

Mehta, C. L.

S. Kumar and C. L. Mehta, Phys. Rev. A 21, 1573 (1980);Phys. Rev. A 24, 1460 (1981).
[Crossref]

Milburn, G. J.

G. J. Milburn, J. Phys. (Paris) A 17, 737 (1984).

Mollow, B. R.

B. R. Mollow and R. J. Glauber, Phys. Rev. 160, 1076 (1976).
[Crossref]

B. R. Mollow and R. J. Glauber, Phys. Rev. 160, 1097 (1976).
[Crossref]

Narozhny, N. B.

N. B. Narozhny, J. J. Sanchez-Mondragon, and J. H. Eberly, Phys. Rev. A 23, 236 (1981), and references therein.
[Crossref]

Parker, L.

L. Parker, Phys. Rev. 183, 1057 (1969).
[Crossref]

Radmore, P. M.

P. L. Knight and P. M. Radmore, Phys. Rev. A 26, 676 (1982).
[Crossref]

P. L. Knight and P. M. Radmore, Phys. Lett. 90A, 342 (1982).

Sanchez-Mondragon, J. J.

N. B. Narozhny, J. J. Sanchez-Mondragon, and J. H. Eberly, Phys. Rev. A 23, 236 (1981), and references therein.
[Crossref]

Siegman, A. E.

W. H. Louisell, A. Yariv, and A. E. Siegman, Phys. Rev. 124, 1646 (1961).
[Crossref]

Takahashi, Y.

Y. Takahashi and H. Umezawa, Collect. Phenom. 2, 55 (1975).

Ter Haar, D.

We have used the thermal-density matrix ρth= Z−1(β) e−βH.See, for example, the review by D. Ter Haar, Rep. Prog. Phys. 24, 304 (1961).
[Crossref]

Umezawa, H.

Y. Takahashi and H. Umezawa, Collect. Phenom. 2, 55 (1975).

Walls, D. F.

D. F. Walls, Nature 306, 141 (1983).
[Crossref]

Yariv, A.

W. H. Louisell, A. Yariv, and A. E. Siegman, Phys. Rev. 124, 1646 (1961).
[Crossref]

Yuen, H. P.

H. P. Yuen, Phys. Rev. A 13, 2226 (1976).
[Crossref]

Collect. Phenom. (3)

Y. Takahashi and H. Umezawa, Collect. Phenom. 2, 55 (1975).

H. J. Kreuzer and C. G. Kuper, Collect. Phenom. 2, 141 (1976).

H. J. Kreuzer and C. G. Kuper, Collect. Phenom. 2, 131 (1976);J. Phys. G 2, 9 (1976).

J. Phys. (Paris) A (1)

G. J. Milburn, J. Phys. (Paris) A 17, 737 (1984).

Nature (2)

D. F. Walls, Nature 306, 141 (1983).
[Crossref]

S. W. Hawking, Nature 248, 30 (1974);Commun. Math. Phys. 43, 199 (1975);P. C. W. Davies, J. Phys. (Paris) A 8, 609 (1975);Rep. Prog. Phys. 41, 1313 (1978);W. G. Unruh, Phys. Rev. D 14, 870 (1976).
[Crossref]

Opt. Acta (1)

S. M. Barnett and P. L. Knight, Opt. Acta 31, 435 (1984), and references therein.
[Crossref]

Phys. Lett. (2)

P. L. Knight and P. M. Radmore, Phys. Lett. 90A, 342 (1982).

W. Israel, Phys. Lett. 57A, 107 (1976).

Phys. Rev. (6)

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

L. Parker, Phys. Rev. 183, 1057 (1969).
[Crossref]

W. H. Louisell, A. Yariv, and A. E. Siegman, Phys. Rev. 124, 1646 (1961).
[Crossref]

B. R. Mollow and R. J. Glauber, Phys. Rev. 160, 1076 (1976).
[Crossref]

B. R. Mollow and R. J. Glauber, Phys. Rev. 160, 1097 (1976).
[Crossref]

G. Lachs, Phys. Rev. 138, B1012 (1965).
[Crossref]

Phys. Rev. A (4)

P. L. Knight and P. M. Radmore, Phys. Rev. A 26, 676 (1982).
[Crossref]

N. B. Narozhny, J. J. Sanchez-Mondragon, and J. H. Eberly, Phys. Rev. A 23, 236 (1981), and references therein.
[Crossref]

H. P. Yuen, Phys. Rev. A 13, 2226 (1976).
[Crossref]

S. Kumar and C. L. Mehta, Phys. Rev. A 21, 1573 (1980);Phys. Rev. A 24, 1460 (1981).
[Crossref]

Proc. IEEE (1)

E. T. Jaynes and F. W. Cummings, Proc. IEEE 51, 89 (1963).
[Crossref]

Rep. Prog. Phys. (1)

We have used the thermal-density matrix ρth= Z−1(β) e−βH.See, for example, the review by D. Ter Haar, Rep. Prog. Phys. 24, 304 (1961).
[Crossref]

Other (4)

W. H. Louisell demonstrated the use of two-mode expectation values to evaluate thermal averages of single-mode Bose operators in Quantum Statistical Properties of Radiation (Wiley, New York, 1973), p. 161.

See, for example, W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1973), p. 418;H. Haken, Light (North-Holland, Amsterdam, 1981), Vol. 1, p. 285.

W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1973), p. 336.

If the P function is moresingular than a δ function, we say that it ceases to exist. This is the criterion used by Mollow and Glauber.18,19

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Figures (1)

Fig. 1
Fig. 1

The ladder of states for an initially (a) excited, (b) deexcited atom in a thermofield vacuum |0(β)〉. Note that all the transitions are reversible.

Equations (196)

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A = Z 1 ( β ) Tr { A e β H } ,
Z ( β ) = Tr { e β H } ,
A = 0 ( β ) | A | 0 ( β ) ,
0 ( β ) | A | 0 ( β ) = Z 1 ( β ) n n | A | n e β E n ,
H | n = E n | n ,
n | m = δ n m .
| 0 ( β ) = n | n f n ( β ) .
m n f n * ( β ) n | A | m f m ( β ) = Z 1 ( β ) n n | A | n e β E n ,
f n * ( β ) f m ( β ) = Z 1 ( β ) e β E n δ n m ,
H | ñ = E n | ñ ,
ñ | m = δ ñ m .
f n ( β ) = | ñ exp ( β E n / 2 ) Z 1 / 2 ( β )
| 0 ( β ) = Z 1 / 2 ( β ) n exp ( β E n / 2 ) | n , ñ ,
| n , ñ = | n | ñ .
0 ( β ) | A | 0 ( β ) = Z 1 ( β ) n m exp ( β E n / 2 ) exp ( β E m / 2 ) × n , ñ | A | m , m = Z 1 ( β ) n exp ( β E n ) n | A | n .
H = ω a + a ,
[ a , a + ] = 1 .
H = ω ã ã ,
[ ã , ã + ] = 1
[ a , ã ] = 0 = [ a , ã + ] .
| 0 ( β ) = Z 1 / 2 ( β ) n exp ( β n ω / 2 ) | n , ñ = Z 1 / 2 ( β ) n exp ( β n ω / 2 ) 1 n ! ( a + ) n ( ã + ) n | 0 , 0 = ( 1 e β ω ) 1 / 2 exp { exp ( β ω / 2 ) a + ã + } | 0 , 0 .
| 0 ( β ) = e i G | 0 , 0 ,
G = i θ ( β ) [ ã a a + ã + ] .
G = θ ( β ) [ ã a + a + ã + ] .
a ( β ) = e i G a e i G = u ( β ) a υ ( β ) ã + ,
ã ( β ) = e i G ã e i G = u ( β ) ã υ ( β ) a + ,
u ( β ) = cosh θ ( β ) = ( 1 e β ω ) 1 / 2 ,
υ ( β ) = sinh θ ( β ) = ( e β ω 1 ) 1 / 2 .
a ( β ) = e i G a ( β ) e i G = u ( β ) a ( β ) + υ ( β ) ã + ( β ) ,
ã = e i G ã ( β ) e i G = u ( β ) ã ( β ) + υ ( β ) a + ( β ) ,
a ( β ) | 0 ( β ) = 0 = ã ( β ) | 0 ( β ) ,
{ | n , m , ( β ) } = { e i G | n , m } = { 1 n ! 1 m ! [ a + ( β ) ] n ( ã ( β ) ) m | 0 ( β ) } .
T ( θ ) = e i G ,
T ( θ ) = exp [ θ ( β ) ( ã a a + ã + ) ] .
T ( θ ) = exp { [ θ ( β ) ( ã a θ * ( β ) a + ã + ) ] } .
u ( β ) = cosh χ ,
υ ( β ) = e i ɛ sinh χ .
| α ( β ) , φ ( β ) = exp [ α a + ( β ) α * a ( β ) ] × exp [ φ ã ( β ) φ * ã ( β ) ] | 0 ( β ) .
exp [ α a + ( β ) α * a ( β ) ] = T ( θ ) exp [ α a + α * a ] T 1 ( θ ) = T ( θ ) D ( α ) T 1 ( θ ) ,
| α ( β ) , φ ˆ ( β ) = T ( θ ) D ( α ) D ( φ ) | 0 , 0 ,
D ( φ ) = exp ( φ ã + φ * ã ) .
| α ( β ) , φ ( β ) = D ( α ) D ( φ ) T ( θ ) | 0 , 0 = T ( θ ) D [ u ( β ) α υ ( β ) φ * ] × D [ u ( β ) φ υ ( β ) α * ] | 0 , 0 .
α ( β ) , φ ( β ) | A ( a , a + ) | α ( β ) , φ ( β ) = 0 , 0 | T 1 ( θ ) D 1 ( α ) D 1 ( φ ) A ( a , a + ) × D ( φ ) D ( α ) T ( θ ) | 0 , 0 = 0 , 0 | T 1 ( θ ) D 1 ( α ) A ( a , a + ) D ( α ) T ( θ ) | 0 , 0 ,
α ( β ) , φ ( β ) | A ( a , a + ) | α ( β ) , φ ( β ) = Tr { D 1 ( α ) A ( a , a + ) D ( α ) ρ th } = Tr { A ( a , a + ) D ( α ) ρ th D 1 ( α ) } ,
α ( β ) , φ ( β ) | Ã ( ã , ã + ) | α ( β ) , φ ( β ) = Tr { ã ( ã , ã + ) D ( φ ) ρ th D 1 ( φ ) } .
α ( β ) , φ ( β ) | A ( a , a + ) | α ( β ) , φ ( β ) = Tr { A ( a , a + ) D [ u ( β ) α + υ ( β ) ] φ * ) × ρ th D 1 [ u ( β ) α + υ ( β ) φ * ] } ,
α ( β ) , φ ( β ) | Ã ( ã , ã + ) | α ( β ) , φ ( β ) = Tr { Ã ( ã , ã + ) D [ u ( β ) φ + υ ( β ) α * ] × ρ th D 1 [ u ( β ) φ + υ ( β ) α * ] } .
b , c , | Tr { ρ th a A ( t ) } | b , c , = b , c , | a 0 ( β ) | e i H t A ( 0 ) e i H t | 0 ( β ) a | b , c , ,
d · E = ( λ 1 σ + + λ 1 * σ ) ( λ 2 a + + λ 2 * a ) ,
d · E g ( a + σ + σ + a ) .
H 1 = ½ ω 0 σ 3 + ω a + a + g ( a + σ + σ + a ) .
A ( t ) = ± | 0 ( β ) | e i H t A ( 0 ) e i H t | 0 ( β ) | ± ,
H 2 = ½ ω 0 σ 3 + ω ( a + a ã ã ) + g ( a + σ + σ + a ) .
H 1 = ½ ω 0 σ 3 + g ( e i ω t a + σ + e i ω t σ + a ) .
H 2 = ½ ω 0 σ 3 + ω [ a + ( β ) a ( β ) ã + ( β ) ã ( β ) ] + g u ( β ) [ a + ( β ) σ + σ + a ( β ) ] + g υ ( β ) [ ã + ( β ) σ + σ + ã + ( β ) ] .
H 2 T 1 ( θ ) H 2 T ( θ ) = ½ ω 0 σ 3 + ω ( a + a ã + ã ) + g cosh θ ( a + σ + σ + a ) + g sinh θ ( ã σ + σ + ã + ) ,
| 0 ( β ) T 1 ( θ ) | 0 ( β ) = | 0 , 0 .
A T 1 ( θ ) A T ( θ )
V 1 = g u ( β ) [ a + ( β ) σ + σ + a ( β ) ] ,
V 2 = g υ ( β ) [ ã ( β ) σ + σ + ã + ( β ) ] .
+ | n , ñ , ( β ) | V 1 | n + 1 , ñ , ( β ) | = g u ( β ) n + 1 ,
+ | n , ñ , ( β ) | V 2 | n , ñ 1 , ( β ) | = g υ ( β ) n ,
| n , m , ( β ) [ a + ( β ) ] n n ! [ ã + ( β ) ] m m ! | 0 ( β ) .
τ c ( g 1 ) 1 = 1 / g ,
τ c [ g υ ( β ) 1 ] 1 = 1 g n ¯ ,
u ( β ) = ( n ¯ + 1 ) 1 / 2 υ ( β ) = n ¯ 1 / 2 ,
u ( β ) = ( n ¯ + 1 ) 1 / 2 υ ( β ) = n ¯ 1 / 2 ,
H 2 I = V = g u ( β ) [ a + ( β ) σ + σ + a ( β ) ] + g υ ( β ) [ ã ( β ) σ + σ + ã + ( β ) ] .
d d t σ + = i g [ u ( β ) a + ( β ) + υ ( β ) ã ( β ) ] σ 3 ,
d d t a + ( β ) = i g u ( β ) σ + ,
d d t ã + ( β ) = i g u ( β ) σ ,
d d t σ + σ = i g [ u ( β ) a + ( β ) + υ ( β ) ã ( β ) ] σ i g σ + [ u ( β ) a ( β ) + υ ( β ) ã + ( β ) ] ,
d d t a + ( β ) σ = i g u ( β ) σ + σ + i g [ u ( β ) a + ( β ) a ( β ) + υ ( β ) a + ( β ) ã + ( β ) ] σ 3 ,
d d t ã ( β ) σ = i g u ( β ) σ + σ + i g [ υ ( β ) ã ( β ) ã + ( β ) + u ( β ) ã ( β ) a ( β ) ] σ 3 ,
d 2 d t 2 σ + σ = 2 V 2 σ 3
d d t 2 σ 3 = 4 V 2 σ 3 ,
V 2 = g 2 { [ u 2 ( β ) a + ( β ) a ( β ) + u ( β ) υ ( β ) a + ( β ) ã + ( β ) + u ( β ) υ ( β ) ã ( β ) a ( β ) + υ 2 ( β ) ã ( β ) ã + ( β ) ] + σ + σ } .
1 g 2 V 2 M = σ + σ + a + ( β ) a ( β ) ã + ( β ) ã ( β ) .
H 0 = ω a + a ,
H 0 = ω ã + ã .
H int = U ( t ) θ ( t ) { a + ã + ã a } ,
a ( t ) = e i Ω t cosh Φ ( t ) a ( 0 ) + e i Ω t sinh Φ ( t ) ã + ( 0 ) ,
Ω = ½ ( ω ω ) + ½ { ( ω ω ) 2 4 U 2 } 1 / 2 ,
Ω = ½ ( ω ω ) + ½ { ( ω ω ) 2 4 U 2 } 1 / 2 ,
tanh 2 Φ ( t ) = 2 U ( t ) / ( ω + ω ) .
n ( t ) = 2 U 2 ( ω + ω ) 2 4 U 2 ( 1 cos { [ ( ω + ω ) 2 4 U 2 ] 1 / 2 t } ) .
U ( t ) ½ [ ω + ω ( t ) ] e β ω / 2 / [ 1 + e β ω ] ,
ω ( t ) ω + π / [ ( 1 e β ω ) 1 / 2 t ] ,
n ( e β ω 1 ) 1 ,
H 1 = ω a + a + j ω j b j + b j + j ( K j a + b j + K j * b j + a ) .
H 2 = ω a + a + j ω j ( b j + b j b j + b j ) + j ( K j a + b j + K j * b j + a ) .
H 2 = ω a + a + j ω j [ b j + ( β ) b j ( β ) b j + ( β ) b j ( β ) ] + j [ u j K j a + b j ( β ) + ( β ) + u j K j * b j + ( β ) a ] + j [ υ j K j a + b j + ( β ) + υ j K j * b j ( β ) a ] ,
u j = ( 1 e β ω j ) 1 / 2 ,
υ j = ( e β ω j 1 ) 1 / 2 .
d d t a = i ω a i j K j u j b j ( β ) i j K j υ j b j + ( β ) ,
d d t b j ( β ) = i ω j b j ( β ) i K j * u j a ,
d d t b j + ( β ) = i ω j b j + ( β ) i K j * υ j a ,
d d t a = i ω a i j | K j | 2 u j 2 0 t a ( t ) exp [ i ω j ( t t ) ] d t + j | K j | 2 υ j 2 0 t a ( t ) exp [ i ω j ( t t ) ] d t + G + F + ,
G = i j K j u j b j ( 0 ) ( β ) exp ( i ω j t ) ,
F + = i j K j υ j b j + ( 0 ) ( β ) exp ( i ω j t ) .
u j 2 υ j 2 = 1
d d t a = i ω a + j | K j | 2 0 t a ( t ) × exp [ i ω j ( t t ) ] d t + G + F + .
F ( t 1 ) F + ( t 2 n ) G ( t 2 n + 1 ) G + ( t 2 n + 2 M ) = F ( t 1 ) F + ( t 2 n ) G ( t 2 n + 1 ) G + ( t 2 n + 2 M ) ,
G ( t ) G + ( t ) = j | K j | 2 u j 2 exp [ i ω j ( t t ) ] = j | K j | 2 ( n ¯ j + 1 ) exp [ i ω j ( t t ) ] .
F ( t ) F + ( t ) = j | K j | 2 υ j 2 exp [ i ω j ( t t ) ] = j | K j | 2 n ¯ j exp [ i ω j ( t t ) ] .
d d t a + a = γ a + a + γ n ¯ .
s t = i , j δ ( ω i ω j ) { [ Q i Q j s Q j s Q i ] W i j + [ Q i s Q j s Q j Q i ] W i j ,
Q 1 = a + ,
Q 2 = a .
W i j + = 0 exp ( i ω i τ ) F i ( τ ) F j d τ ,
W i j = 0 exp ( i ω i τ ) F j F i ( τ ) d τ ,
F 1 = j K j b j ,
F 2 = j K j * b j + .
G 1 = j K j u j b j ( β ) ,
G 2 = j K j * u j b j + ( β ) ,
F 1 = j K j * υ j b j ( β ) ,
F 2 = j K j υ j b j + ( β ) .
F 1 = G 1 + F 2 ,
F 2 = G 2 + F 1
s t = γ 2 ( n ¯ + 1 ) [ 2 a s a + a + a s s a + a ] + γ 2 n ¯ [ 2 a + s a a a + s s a a + ] .
H = ω a a + a + ω b b + b i { g a b e i ω t g * b + a + e i ω t } ,
H I = i { g a b g * b + a + } .
exp ( i H I t ) = exp [ ( g t a b g * t b + a + ) ]
exp ( i H I t ) = T ( g t ) .
| ψ = Z 1 / 2 ( β ) n exp ( β n ω / 2 ) | n a | x ( n ) b ,
| ψ = Z 1 / 2 ( β ) n exp ( β n ω / 2 ) | y ( n ) a | n b ,
| ψ = Z 1 / 2 ( β ) n exp ( β n ω / 2 ) | n a | n b = | 0 ( β ) .
[ ( a + a b + b ) , H ] = 0 .
T ( θ ) [ H H ] T 1 ( θ ) = H H .
| t = T ( g t ) | ,
a ( t ) = T 1 ( g t ) a ( 0 ) T ( g t ) = a ( 0 ) cosh g t + b ( 0 ) sinh g t ,
b ( t ) = T 1 ( g t ) b ( 0 ) T ( g t ) = b ( 0 ) cosh g t + a ( 0 ) sinh g t .
| t = exp [ g t ( a b b + a + ) ] | 0 a , 0 b ,
n ¯ ( t ) = sinh 2 g t .
| = exp ( α a + α * a ) exp ( ξ b + ξ * b ) | 0 a , 0 b ,
| t = exp [ g t ( a b b + a + ) ] exp ( α a + α * a ) × exp ( ξ b + ξ * b ) | 0 a , a b ,
n ¯ a = sinh 2 g t + | α cosh g t + ξ * sinh g t | 2 ,
n ¯ b = sinh 2 g t + | ξ cosh g t + α * sinh g t | 2 ,
α a ( t ) = α cosh g t + ξ * sinh g t ,
α b ( t ) = ξ cosh g t + α * sinh g t .
lim t | α a ( t ) | 2 n ¯ a ( t ) = | α + ξ * | 2 1 + | α + ξ * | 2 = lim t | α b ( t ) | 2 n ¯ b ( t ) .
ρ A ( 0 ) = d 2 α P ( α ) | α α | ,
ρ B ( 0 ) = d 2 ξ P ( ξ ) | ξ ξ | ,
U ( η ) = exp ( η a + b η * b + a ) ,
c = U ( η ) a U 1 ( η ) = a cos μ b sin μ e i ϕ ,
a = U ( η ) b U 1 ( η ) = b cos μ + a sin μ e i ϕ .
a = U 1 ( η ) a 1 U ( η ) = a 1 cos μ + b 1 sin μ e i ϕ ,
b = U 1 ( η ) b 1 U ( η ) = b 1 cos μ a 1 sin μ e i ϕ .
[ ( a + a + b + b ) , ( η a + b + η * b + a ) ] = 0 .
U ( η ) | 0 a , 0 b = | 0 a , 0 b .
igab + i g b + a + = i g [ a 1 b 1 ( cos 2 μ sin 2 μ ) + sin μ cos μ ( b 1 2 e i ϕ a 1 2 e i ϕ ) ] + i g [ a 1 + b 1 + ( cos 2 μ sin 2 μ ) + sin μ cos μ ( b 1 + 2 e i ϕ a 1 + 2 e i ϕ ) ] .
H I = i g 2 ( e i ϕ a 1 2 + e i ϕ a 1 + 2 ) i g 2 ( e i ϕ b 1 2 e i ϕ b 1 + 2 ) .
exp ( i H I t ) = exp [ g t 2 ( e i ϕ a 1 2 e i ϕ a 1 + 2 ) ] × exp [ g t 2 ( e i ϕ b 1 2 e i ϕ b 1 + 2 ) ] = S a 1 ( g t e i ϕ ) S b 1 ( g t e i ϕ ) ,
a 1 = X ¯ 1 + i X ¯ 2 ,
b 1 = Y ¯ 1 + i Y ¯ 2 ,
[ Z ¯ 1 , Z ¯ 2 ] = i / 2 ,
Δ Z ¯ 1 Δ Z ¯ 2 1 / 4 .
Δ Z ¯ k 2 < 1 / 4 ,
X ¯ k ( t ) = S a 1 1 ( g t ) S b 1 1 ( g t ) X ¯ k ( 0 ) S a 1 ( g t ) S b 1 ( g t ) = X ¯ k ( 0 ) exp [ ( 1 ) k g t ] ,
Y ¯ k ( t ) = S a 1 1 ( g t ) S b 1 1 ( g t ) Y ¯ k ( 0 ) S a 1 ( g t ) S b 1 ( g t ) = Y ¯ k ( 0 ) exp [ ( 1 ) k + 1 g t ] .
Δ Z ¯ k 2 = Z ¯ k 2 Z ¯ k 2 .
Δ X ¯ k 2 ( t ) = Δ X ¯ k 2 ( 0 ) exp [ ( 1 ) k 2 g t ] ,
Δ Y ¯ k 2 ( t ) = Δ Y ¯ k 2 ( 0 ) exp [ ( 1 ) k + 1 2 g t ] .
| = | 0 a , 0 b ,
| = | 0 a 1 , 0 b 1 ,
Δ X ¯ k 2 ( t ) = ¼ exp [ ( 1 ) k 2 g t ] ,
Δ Y ¯ k 2 ( t ) = ¼ exp [ ( 1 ) k + 1 2 g t ] ,
ρ ( 0 ) = P ( α , ξ ) | α , ξ α , ξ | d 2 α d 2 ξ ,
P ( α , ξ ) = 1 π 2 n 2 exp { | α | 2 n | ξ | 2 n } .
| α ¯ | 2 + | ξ ¯ | 2 = | α | 2 + | ξ | 2 .
ρ ¯ ¯ ( 0 ) = J d 2 α ¯ ¯ d 2 ξ ¯ | α ¯ ¯ , ξ ¯ α ¯ ¯ , ξ ¯ | 1 π 2 n 2 exp { | α ¯ | 2 n | ξ ¯ | 2 n } .
| = exp [ θ ( ã a a + ã + ) ] × exp [ θ ( b b b + b + ) ] | 0 , 0 a | 0 , 0 b .
| = exp [ θ ( ã 1 a 1 + b 1 b 1 ) + θ ( a 1 + ã 1 + + b 1 + b 1 + ) ] × | 0 , 0 a 1 | 0 , 0 b 1 = exp [ θ ( ã 1 a 1 a 1 + ã 1 + ) ] exp [ θ ( b 1 b 1 b 1 + b 1 + ) ] × | 0 , 0 a 1 | 0 , 0 b 1 .
Δ Z ¯ k 2 ( 0 ) = ¼ ( 2 n + 1 ) .
Δ X ¯ k 2 ( t ) = ¼ ( 2 n + 1 ) exp [ ( 1 ) k 2 g t ] ,
Δ Y ¯ k 2 ( t ) = ¼ ( 2 n + 1 ) exp [ ( 1 ) k + 1 2 g t ] .
¼ ( 2 n + 1 ) e 2 g τ = ¼
τ = ln ( 2 n + 1 ) / 2 g .
ρ c ( 0 ) = D a ( α ) D b ( ξ ) ρ ( 0 ) D b 1 ( ξ ) D a 1 ( α ) ,
ρ ¯ c ( 0 ) = D a 1 ( α ¯ ) D b 1 ( ξ ¯ ) ρ ¯ ( 0 ) D b 1 1 ( ξ ¯ ) D a 1 1 ( α ¯ ) ,
α ¯ = 1 2 ( α ξ ) ,
ξ ¯ = 1 2 ( ξ α ) .
ρ ¯ c ( t ) = S a 1 ( g t ) S b 1 ( g t ) D a 1 ( α ¯ ) D b 1 ( ξ ¯ ) ρ ¯ ( 0 ) D b 1 1 ( ξ ¯ ) × D a 1 1 ( α ¯ ) S b 1 1 ( g t ) S a 1 1 ( g t ) = D a 1 [ α ¯ ( t ) ] D b 1 [ ξ ¯ ( t ) ] S a 1 ( g t ) S b 1 ( g t ) ρ ¯ ( 0 ) S b 1 1 ( g t ) × S a 1 1 ( g t ) D b 1 1 [ ξ ¯ ( t ) ] D a 1 1 [ α ¯ ( t ) ] ,
α ¯ ( t ) = α ¯ cosh g t α ¯ * sinh g t ,
ξ ¯ ( t ) = ξ ¯ cosh g t ξ ¯ * sinh g t .
V = i g ( c + b a a + b + c )
d 2 d t 2 N ̂ c = 2 g 2 [ 3 N ̂ c 2 ( 2 Â + 2 B ̂ 1 ) N ̂ c + Â B ̂ ] ,
N ̂ c = c + c ,
 = c + c + a + a ,
B ̂ = c + c + b + b .
V = i g N c 1 / 2 ( t ) [ b a exp ( i ω c t ) a + b + exp ( i ω c t ) ] .
H I = i g N c 1 / 2 ( t ) ( b a a + b + ) .
exp ( i H I t ) = exp [ g ( b a a + b + ) 0 t N c 1 / 2 ( t ) d t ] = T [ g 0 t N c 1 / 2 ( t ) d t ] .
N c 1 / 2 ( t ) a + ( t ) a ( t ) , b + ( t ) b ( t ) ,
| 0 ( β ) = z 1 / 2 ( β ) n exp ( β n ω / 2 ) | n , ñ
| 0 ( β ) | 0 ( β ) | 2 = [ 1 exp ( β ω ) ] [ 1 exp ( β ω ) ] { n = 0 exp ( [ β + β ] n ω / 2 } = cosh [ ( β + β ) ω / 2 ] cosh [ ( β + β ) ω / 2 ] cosh [ ( β + β ) ω / 2 ] 1 1 .

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