Abstract

We propose a stochastic model for continuous photon counting and heterodyne measurement of a coherent source. A nonlinear filtering equation for the posterior state of a single-mode field in a cavity is derived by using the methods of the quantum stochastic calculus. The posterior dynamics is found for the observation of a Bose field being initially in a coherent state. The filtering equations for the counting and diffusion processes are given.

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  1. V. P. Belavkin, “A continuous counting observation and posterior quantum dynamics,” J. Phys. A 22, L1109–L1114 (1989).
    [CrossRef]
  2. V. P. Belavkin, “A posterior Schrödinger equation for continuous nondemolition measurement,” J. Math. Phys. 31, 2930–2934(1990).
    [CrossRef]
  3. A. Barchielli and V. P. Belavkin, “Measurements continuous in time and a posteriori states in quantum mechanics,” J. Phys. A: Math. Gen. 24, 1495–1514 (1991).
    [CrossRef]
  4. V. P. Belavkin, “Quantum stochastic calculus and quantum nonlinear filtering,” J. Multivar. Anal. 42, 171–201 (1992).
    [CrossRef]
  5. V. P. Belavkin and P. Staszewski, “Nondemolition observation of a free quantum particle,” Phys. Rev. A 45, 1347–1356 (1992).
    [CrossRef] [PubMed]
  6. V. P. Belavkin, “Measurement, filtering and control in quantum open dynamical systems,” Rep. Math. Phys. 43, A405–A425(1999).
    [CrossRef]
  7. L. Bouten, M. I. Guţă, and H. Maassen, “Stochastic Schrödinger equations,” J. Phys. A 37, 3189–3209 (2004).
    [CrossRef]
  8. H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control (Cambridge University Press, 2010).
  9. R. L. Hudson and K. R. Parthasarathy, “Quantum Ito’s formula and stochastic evolutions,” Commun. Math. Phys. 93, 301–323(1984).
    [CrossRef]
  10. C. W. Gardiner and M. J. Collet, “Input and output in damped quantum systems: quantum stochastic differential equations and the master equation,” Phys. Rev. A 31, 3761–3774 (1985).
    [CrossRef] [PubMed]
  11. K. R. Parthasarathy, An Introduction to Quantum Stochastic Calculus (Birkhäuser Verlag, 1992).
    [CrossRef]
  12. C. W. Gardiner, Quantum Noise (Springer, 2000).
  13. O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics II (Springer, 1981).
  14. H. M. Wiseman and G. J. Milburn, “Quantum theory of field-quadrature measurements,” Phys. Rev. A 47, 642–662 (1993).
    [CrossRef] [PubMed]
  15. H. M. Wiseman and G. J. Milburn, “Interpretation of quantum jump and diffusion processes illustrated on the Bloch sphere,” Phys. Rev. A 47, 1652–1666 (1993).
    [CrossRef] [PubMed]
  16. H. Carmichael, An Open Systems Approach to Quantum Optics (Springer, 1993).
  17. A. Barchielli, “Direct and heterodyne detection and other applications of quantum stochastic calculus to quantum optics,” Quantum Opt. 2, 423–441 (1990).
    [CrossRef]
  18. A. Barchielli, “Continual measurements in quantum mechanics and quantum stochastic calculus,” in Open Quantum Systems III. Recent Developments, A.Atall, A.Joye, and C.A.Pillet, eds. (Springer LNM, 2006), pp. 207–288.
    [CrossRef]
  19. A. Barchielli and N. Pero, “A quantum stochastic approach to the spectrum of a two-level atom,” J. Opt. B: Quantum Semiclass. Opt. 4, 272–282 (2002).
    [CrossRef]
  20. A. Barchielli, “Measurement theory and stochastic differential equations in quantum mechanics,” Phys. Rev. A 34, 1642–1649(1986).
    [CrossRef] [PubMed]
  21. M. J. Collet, R. Loudon, and C. W. Gardiner, “Quantum theory of optical homodyne and heterodyne detection,” J. Mod. Optics 34, 881–902 (1987).
    [CrossRef]
  22. V. P. Belavkin and S. Edwards, “Quantum filtering and optimal control,” in Quantum Filtering and Optimal Control, V.P.Belavkin and M.Guţă, eds. (World Scientific, 2008), pp. 143–205.

2004 (1)

L. Bouten, M. I. Guţă, and H. Maassen, “Stochastic Schrödinger equations,” J. Phys. A 37, 3189–3209 (2004).
[CrossRef]

2002 (1)

A. Barchielli and N. Pero, “A quantum stochastic approach to the spectrum of a two-level atom,” J. Opt. B: Quantum Semiclass. Opt. 4, 272–282 (2002).
[CrossRef]

1999 (1)

V. P. Belavkin, “Measurement, filtering and control in quantum open dynamical systems,” Rep. Math. Phys. 43, A405–A425(1999).
[CrossRef]

1993 (2)

H. M. Wiseman and G. J. Milburn, “Quantum theory of field-quadrature measurements,” Phys. Rev. A 47, 642–662 (1993).
[CrossRef] [PubMed]

H. M. Wiseman and G. J. Milburn, “Interpretation of quantum jump and diffusion processes illustrated on the Bloch sphere,” Phys. Rev. A 47, 1652–1666 (1993).
[CrossRef] [PubMed]

1992 (2)

V. P. Belavkin, “Quantum stochastic calculus and quantum nonlinear filtering,” J. Multivar. Anal. 42, 171–201 (1992).
[CrossRef]

V. P. Belavkin and P. Staszewski, “Nondemolition observation of a free quantum particle,” Phys. Rev. A 45, 1347–1356 (1992).
[CrossRef] [PubMed]

1991 (1)

A. Barchielli and V. P. Belavkin, “Measurements continuous in time and a posteriori states in quantum mechanics,” J. Phys. A: Math. Gen. 24, 1495–1514 (1991).
[CrossRef]

1990 (2)

V. P. Belavkin, “A posterior Schrödinger equation for continuous nondemolition measurement,” J. Math. Phys. 31, 2930–2934(1990).
[CrossRef]

A. Barchielli, “Direct and heterodyne detection and other applications of quantum stochastic calculus to quantum optics,” Quantum Opt. 2, 423–441 (1990).
[CrossRef]

1989 (1)

V. P. Belavkin, “A continuous counting observation and posterior quantum dynamics,” J. Phys. A 22, L1109–L1114 (1989).
[CrossRef]

1987 (1)

M. J. Collet, R. Loudon, and C. W. Gardiner, “Quantum theory of optical homodyne and heterodyne detection,” J. Mod. Optics 34, 881–902 (1987).
[CrossRef]

1986 (1)

A. Barchielli, “Measurement theory and stochastic differential equations in quantum mechanics,” Phys. Rev. A 34, 1642–1649(1986).
[CrossRef] [PubMed]

1985 (1)

C. W. Gardiner and M. J. Collet, “Input and output in damped quantum systems: quantum stochastic differential equations and the master equation,” Phys. Rev. A 31, 3761–3774 (1985).
[CrossRef] [PubMed]

1984 (1)

R. L. Hudson and K. R. Parthasarathy, “Quantum Ito’s formula and stochastic evolutions,” Commun. Math. Phys. 93, 301–323(1984).
[CrossRef]

Barchielli, A.

A. Barchielli and N. Pero, “A quantum stochastic approach to the spectrum of a two-level atom,” J. Opt. B: Quantum Semiclass. Opt. 4, 272–282 (2002).
[CrossRef]

A. Barchielli and V. P. Belavkin, “Measurements continuous in time and a posteriori states in quantum mechanics,” J. Phys. A: Math. Gen. 24, 1495–1514 (1991).
[CrossRef]

A. Barchielli, “Direct and heterodyne detection and other applications of quantum stochastic calculus to quantum optics,” Quantum Opt. 2, 423–441 (1990).
[CrossRef]

A. Barchielli, “Measurement theory and stochastic differential equations in quantum mechanics,” Phys. Rev. A 34, 1642–1649(1986).
[CrossRef] [PubMed]

A. Barchielli, “Continual measurements in quantum mechanics and quantum stochastic calculus,” in Open Quantum Systems III. Recent Developments, A.Atall, A.Joye, and C.A.Pillet, eds. (Springer LNM, 2006), pp. 207–288.
[CrossRef]

Belavkin, V. P.

V. P. Belavkin, “Measurement, filtering and control in quantum open dynamical systems,” Rep. Math. Phys. 43, A405–A425(1999).
[CrossRef]

V. P. Belavkin, “Quantum stochastic calculus and quantum nonlinear filtering,” J. Multivar. Anal. 42, 171–201 (1992).
[CrossRef]

V. P. Belavkin and P. Staszewski, “Nondemolition observation of a free quantum particle,” Phys. Rev. A 45, 1347–1356 (1992).
[CrossRef] [PubMed]

A. Barchielli and V. P. Belavkin, “Measurements continuous in time and a posteriori states in quantum mechanics,” J. Phys. A: Math. Gen. 24, 1495–1514 (1991).
[CrossRef]

V. P. Belavkin, “A posterior Schrödinger equation for continuous nondemolition measurement,” J. Math. Phys. 31, 2930–2934(1990).
[CrossRef]

V. P. Belavkin, “A continuous counting observation and posterior quantum dynamics,” J. Phys. A 22, L1109–L1114 (1989).
[CrossRef]

V. P. Belavkin and S. Edwards, “Quantum filtering and optimal control,” in Quantum Filtering and Optimal Control, V.P.Belavkin and M.Guţă, eds. (World Scientific, 2008), pp. 143–205.

Bouten, L.

L. Bouten, M. I. Guţă, and H. Maassen, “Stochastic Schrödinger equations,” J. Phys. A 37, 3189–3209 (2004).
[CrossRef]

Bratteli, O.

O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics II (Springer, 1981).

Carmichael, H.

H. Carmichael, An Open Systems Approach to Quantum Optics (Springer, 1993).

Collet, M. J.

M. J. Collet, R. Loudon, and C. W. Gardiner, “Quantum theory of optical homodyne and heterodyne detection,” J. Mod. Optics 34, 881–902 (1987).
[CrossRef]

C. W. Gardiner and M. J. Collet, “Input and output in damped quantum systems: quantum stochastic differential equations and the master equation,” Phys. Rev. A 31, 3761–3774 (1985).
[CrossRef] [PubMed]

Edwards, S.

V. P. Belavkin and S. Edwards, “Quantum filtering and optimal control,” in Quantum Filtering and Optimal Control, V.P.Belavkin and M.Guţă, eds. (World Scientific, 2008), pp. 143–205.

Gardiner, C. W.

M. J. Collet, R. Loudon, and C. W. Gardiner, “Quantum theory of optical homodyne and heterodyne detection,” J. Mod. Optics 34, 881–902 (1987).
[CrossRef]

C. W. Gardiner and M. J. Collet, “Input and output in damped quantum systems: quantum stochastic differential equations and the master equation,” Phys. Rev. A 31, 3761–3774 (1985).
[CrossRef] [PubMed]

C. W. Gardiner, Quantum Noise (Springer, 2000).

Guta, M. I.

L. Bouten, M. I. Guţă, and H. Maassen, “Stochastic Schrödinger equations,” J. Phys. A 37, 3189–3209 (2004).
[CrossRef]

Hudson, R. L.

R. L. Hudson and K. R. Parthasarathy, “Quantum Ito’s formula and stochastic evolutions,” Commun. Math. Phys. 93, 301–323(1984).
[CrossRef]

Loudon, R.

M. J. Collet, R. Loudon, and C. W. Gardiner, “Quantum theory of optical homodyne and heterodyne detection,” J. Mod. Optics 34, 881–902 (1987).
[CrossRef]

Maassen, H.

L. Bouten, M. I. Guţă, and H. Maassen, “Stochastic Schrödinger equations,” J. Phys. A 37, 3189–3209 (2004).
[CrossRef]

Milburn, G. J.

H. M. Wiseman and G. J. Milburn, “Interpretation of quantum jump and diffusion processes illustrated on the Bloch sphere,” Phys. Rev. A 47, 1652–1666 (1993).
[CrossRef] [PubMed]

H. M. Wiseman and G. J. Milburn, “Quantum theory of field-quadrature measurements,” Phys. Rev. A 47, 642–662 (1993).
[CrossRef] [PubMed]

H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control (Cambridge University Press, 2010).

Parthasarathy, K. R.

R. L. Hudson and K. R. Parthasarathy, “Quantum Ito’s formula and stochastic evolutions,” Commun. Math. Phys. 93, 301–323(1984).
[CrossRef]

K. R. Parthasarathy, An Introduction to Quantum Stochastic Calculus (Birkhäuser Verlag, 1992).
[CrossRef]

Pero, N.

A. Barchielli and N. Pero, “A quantum stochastic approach to the spectrum of a two-level atom,” J. Opt. B: Quantum Semiclass. Opt. 4, 272–282 (2002).
[CrossRef]

Robinson, D. W.

O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics II (Springer, 1981).

Staszewski, P.

V. P. Belavkin and P. Staszewski, “Nondemolition observation of a free quantum particle,” Phys. Rev. A 45, 1347–1356 (1992).
[CrossRef] [PubMed]

Wiseman, H. M.

H. M. Wiseman and G. J. Milburn, “Quantum theory of field-quadrature measurements,” Phys. Rev. A 47, 642–662 (1993).
[CrossRef] [PubMed]

H. M. Wiseman and G. J. Milburn, “Interpretation of quantum jump and diffusion processes illustrated on the Bloch sphere,” Phys. Rev. A 47, 1652–1666 (1993).
[CrossRef] [PubMed]

H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control (Cambridge University Press, 2010).

Commun. Math. Phys. (1)

R. L. Hudson and K. R. Parthasarathy, “Quantum Ito’s formula and stochastic evolutions,” Commun. Math. Phys. 93, 301–323(1984).
[CrossRef]

J. Math. Phys. (1)

V. P. Belavkin, “A posterior Schrödinger equation for continuous nondemolition measurement,” J. Math. Phys. 31, 2930–2934(1990).
[CrossRef]

J. Mod. Optics (1)

M. J. Collet, R. Loudon, and C. W. Gardiner, “Quantum theory of optical homodyne and heterodyne detection,” J. Mod. Optics 34, 881–902 (1987).
[CrossRef]

J. Multivar. Anal. (1)

V. P. Belavkin, “Quantum stochastic calculus and quantum nonlinear filtering,” J. Multivar. Anal. 42, 171–201 (1992).
[CrossRef]

J. Opt. B: Quantum Semiclass. Opt. (1)

A. Barchielli and N. Pero, “A quantum stochastic approach to the spectrum of a two-level atom,” J. Opt. B: Quantum Semiclass. Opt. 4, 272–282 (2002).
[CrossRef]

J. Phys. A (2)

L. Bouten, M. I. Guţă, and H. Maassen, “Stochastic Schrödinger equations,” J. Phys. A 37, 3189–3209 (2004).
[CrossRef]

V. P. Belavkin, “A continuous counting observation and posterior quantum dynamics,” J. Phys. A 22, L1109–L1114 (1989).
[CrossRef]

J. Phys. A: Math. Gen. (1)

A. Barchielli and V. P. Belavkin, “Measurements continuous in time and a posteriori states in quantum mechanics,” J. Phys. A: Math. Gen. 24, 1495–1514 (1991).
[CrossRef]

Phys. Rev. A (5)

V. P. Belavkin and P. Staszewski, “Nondemolition observation of a free quantum particle,” Phys. Rev. A 45, 1347–1356 (1992).
[CrossRef] [PubMed]

C. W. Gardiner and M. J. Collet, “Input and output in damped quantum systems: quantum stochastic differential equations and the master equation,” Phys. Rev. A 31, 3761–3774 (1985).
[CrossRef] [PubMed]

H. M. Wiseman and G. J. Milburn, “Quantum theory of field-quadrature measurements,” Phys. Rev. A 47, 642–662 (1993).
[CrossRef] [PubMed]

H. M. Wiseman and G. J. Milburn, “Interpretation of quantum jump and diffusion processes illustrated on the Bloch sphere,” Phys. Rev. A 47, 1652–1666 (1993).
[CrossRef] [PubMed]

A. Barchielli, “Measurement theory and stochastic differential equations in quantum mechanics,” Phys. Rev. A 34, 1642–1649(1986).
[CrossRef] [PubMed]

Quantum Opt. (1)

A. Barchielli, “Direct and heterodyne detection and other applications of quantum stochastic calculus to quantum optics,” Quantum Opt. 2, 423–441 (1990).
[CrossRef]

Rep. Math. Phys. (1)

V. P. Belavkin, “Measurement, filtering and control in quantum open dynamical systems,” Rep. Math. Phys. 43, A405–A425(1999).
[CrossRef]

Other (7)

K. R. Parthasarathy, An Introduction to Quantum Stochastic Calculus (Birkhäuser Verlag, 1992).
[CrossRef]

C. W. Gardiner, Quantum Noise (Springer, 2000).

O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics II (Springer, 1981).

A. Barchielli, “Continual measurements in quantum mechanics and quantum stochastic calculus,” in Open Quantum Systems III. Recent Developments, A.Atall, A.Joye, and C.A.Pillet, eds. (Springer LNM, 2006), pp. 207–288.
[CrossRef]

H. Carmichael, An Open Systems Approach to Quantum Optics (Springer, 1993).

H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control (Cambridge University Press, 2010).

V. P. Belavkin and S. Edwards, “Quantum filtering and optimal control,” in Quantum Filtering and Optimal Control, V.P.Belavkin and M.Guţă, eds. (World Scientific, 2008), pp. 143–205.

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Figures (1)

Fig. 1
Fig. 1

Heterodyne setup.

Equations (63)

Equations on this page are rendered with MathJax. Learn more.

F = C ( k = 1 K s k ) .
e ( f ) = ( 1 , f , ( 2 ! ) 1 / 2 f f , ( 3 ! ) 1 / 2 f f f , ) .
e ( g ) | e ( f ) F = exp g | f K exp [ j = 1 n 0 g j ( t ) ¯ f j ( t ) d t ] .
ι ( f ) = exp ( 1 2 f K 2 ) e ( f ) ,
A j ( t ) e ( f ) = 0 t f j ( s ) d s e ( f ) ,
A j ( t ) e ( f ) = ϵ j e ( f + ϵ χ [ 0 , t ) ) | ϵ = 0 ,
Λ i j ( t ) e ( f ) = i d d λ e ( exp ( i λ P i j χ [ 0 , t ) ) f ) | λ = 0 ,
[ A i ( t ) , A j ( t ) ] = [ A i ( t ) , A j ( t ) ] = 0 , [ Λ i j ( t ) , Λ k l ( t ) ] = δ j k Λ i l ( t t ) δ i l Λ k j ( t t ) , [ A i ( t ) , A j ( t ) ] = δ i j t t , [ A j ( t ) , Λ k l ( t ) ] = δ j k A l ( t t ) , [ Λ k l ( t ) , A j ( t ) ] = δ l j A k ( t t ) ,
F = F [ 0 , t ) F [ t , ) ,
d M ( t ) = j = 1 n ( i = 1 n F j i ( t ) d Λ j i ( t ) + E j ( t ) d A j ( t ) + D j ( t ) d A j ( t ) ) + C ( t ) d t ,
d ( M ( t ) M ( t ) ) = d M ( t ) M ( t ) + M ( t ) d M ( t ) + d M ( t ) d M ( t ) ,
d A i ( t ) d A j ( t ) = δ i j d t , d A i ( t ) d Λ k j ( t ) = δ i k d A j ( t ) , d Λ k j ( t ) d A i ( t ) = δ j i d A k ( t ) , d Λ i j ( t ) d Λ k l ( t ) = δ j k d Λ i l ( t ) ,
d U ( t ) = ( μ b d A ( t ) μ b d A ( t ) μ 2 b b d t i H d t ) U ( t ) , U ( 0 ) = I ,
A out ( t ) = U ( t ) A ( t ) U ( t ) , Λ out ( t ) = U ( t ) Λ ( t ) U ( t ) .
Λ out ( t ) = 0 t ( d Λ ( t ) + μ b t d A ( t ) + μ b t d A ( t ) + μ b t b t d t ) ,
[ Λ out ( t ) , U ( t ) Z U ( t ) ] = 0 0 t t ,
[ Λ out ( t ) , Λ out ( t ) ] = 0     t , t 0.
g ( k , t ) :     Z g ( k , t ) [ Z ] , ψ | g ( k , t ) [ Z ] ψ = ψ ι ( f ) | G out ( k , t ) Z t ψ ι ( f ) ,
G out ( k , t ) = exp { 0 t ln k ( t ) d Λ out ( t ) } ,
d G ( k , t ) = ( k ( t ) 1 ) d Λ ( t ) G ( k , t ) .
d π k out ( t , Z ) = ( K t π k out ( t , Z ) + π k out ( t , Z ) K t ) d t + k ( t ) μ b t π k out ( t , Z ) b t d t + μ ( b t π k out ( t , Z ) π k out ( t , Z ) b t ) d A ( t ) + μ ( π k out ( t , Z ) b t b t π k out ( t , Z ) ) d A ( t ) + ( k ( t ) 1 ) ( π k out ( t , Z ) d Λ ( t ) + μ b t π k out ( t , Z ) d A ( t ) + μ π k out ( t , Z ) b t d A ( t ) ) ,
d π k out ( t , Z ) = η ( t ) | ( K π k ( t , Z ) + π k ( t , Z ) K ) + k ( t ) μ b π k ( t , Z ) b + μ ( b π k ( t , Z ) π k ( t , Z ) b ) f ( t ) + μ ( π k ( t , Z ) b b π k ( t , Z ) ) f ( t ) ¯ + ( k ( t ) 1 ) ( π k ( t , Z ) | f ( t ) | 2 + μ b π k ( t , Z ) f ( t ) + μ π k ( t , Z ) b f ( t ) ¯ ) | η ( t ) d t ,
d d t g ( k , t ) [ Z ] = g ( k , t ) [ K Z Z K μ ( Z b f ( t ) + b Z f ( t ) ¯ ) Z | f ( t ) | 2 + k ( t ) ( μ b Z b + μ b Z f ( t ) + μ Z b f ( t ) ¯ + Z | f ( t ) | 2 ) ] ,
g ( k , t ) [ Z ] = n = 0 0 t d t n 0 t n d t n 1 0 t 2 d t 1 k ( t 1 ) k ( t n ) ×
× S ( t 1 ) S ( t n ) Z ( t ) S ( t n ) S ( t 1 ) ,
Z ( t ) = e L ( t ) Z e L ( t ) ,
S ( t ) = e L ( t ) ( μ b + f ( t ) ) e L ( t ) ,
L ( t ) = K t + 0 t ( μ b f ( t ) + | f ( t ) | 2 2 ) d t .
g ( k , t ) [ Z ] = τ Σ t k ( τ ) V ( τ | t ) Z V ( τ | t ) d τ ,
V ^ ( t ) = e L ( t ) n = 0 0 t d t n 0 t n d t n 1 0 t 2 d t 1 S ( t n ) S ( t 1 ) i = 1 n d N ( t i ) .
S ( t ) = e L ( t ) ( μ b + f ( t ) I ) e L ( t ) ,
d V ^ ( t ) = ( K + μ b f ( t ) + | f ( t ) | 2 2 ) V ^ ( t ) d t + ( μ b + f ( t ) I ) V ^ ( t ) d N ( t ) , V ^ ( 0 ) = I .
d ψ ^ ( t ) = ( K + μ b f ( t ) + | f ( t ) | 2 2 ) ψ ^ ( t ) d t + ( μ b + f ( t ) I ) ψ ^ ( t ) d N ( t ) , ψ ^ ( 0 ) = ψ .
d ( ψ ^ ( t ) | ψ ^ ( t ) ) = ψ ^ ( t ) | ( μ b + f ( t ) ¯ ) ( μ b + f ( t ) ) | ψ ^ ( t ) d t + ψ ^ ( t ) | [ ( μ b + f ( t ) ¯ ) ( μ b + f ( t ) ) 1 ] | ψ ^ ( t ) d N ( t ) ,
d ( ψ ^ ( t ) | ψ ^ ( t ) 1 / 2 ) = ψ ^ ( t ) | ψ ^ ( t ) 1 / 2 { 1 2 ( μ b b t + 2 μ Re ( b t f ( t ) ¯ ) + | f ( t ) | 2 ) d t + [ ( μ b b t + 2 μ Re ( b t f ( t ) ¯ ) + | f ( t ) | 2 ) 1 / 2 1 ] d N ( t ) } ,
d φ ^ ( t ) = d ( ψ ^ ( t ) | ψ ^ ( t ) 1 / 2 ) ψ ^ ( t ) + ψ ^ ( t ) | ψ ^ ( t ) 1 / 2 d ψ ^ ( t ) + d ( ψ ^ ( t ) | ψ ^ ( t ) 1 / 2 ) d ψ ^ ( t ) ,
d φ ^ ( t ) = ( K μ b f ( t ) + μ / 2 b b t + μ Re ( b t f ( t ) ¯ ) ) φ ^ ( t ) d t + ( μ b + f ( t ) μ b b t + 2 μ Re ( b t f ( t ) ¯ ) + | f ( t ) | 2 I ) φ ^ ( t ) d N ( t ) , φ ^ ( 0 ) = ψ .
d ρ ^ ( t ) = i [ H , ρ ^ ( t ) ] d t μ 2 { b b , ρ ^ ( t ) } d t + μ b ρ ^ ( t ) b d t + μ [ b f ( t ) ¯ b f ( t ) , ρ ^ ( t ) ] d t + ( ( μ b + f ( t ) ) ρ ^ ( t ) ( μ b + f ( t ) ¯ ) μ b b t + 2 μ Re ( b t f ( t ) ¯ ) + | f ( t ) | 2 ρ ^ ( t ) ) × ( d N ( t ) μ b b t d t 2 μ Re ( b t f ( t ) ¯ ) d t | f ( t ) | 2 d t ) ,
d d t σ ( t ) = i [ H , σ ( t ) ] μ 2 { b b , σ ( t ) } + μ b σ ( t ) b + μ [ b f ( t ) ¯ b f ( t ) , σ ( t ) ]
σ ( t ) = ρ ^ ( t ) s t ,
d N ( t ) ( τ ) = μ b b t d t + 2 μ Re ( b t f ( t ) ¯ ) d t + | f ( t ) | 2 d t .
A mix ( t ) = T A out ( t ) + i 1 T A lo ( t ) .
G out ( k , t ) = ι ( f lo ) | exp { 0 t ln k ( t ) d A mix ( t ) A ˙ mix ( t ) } ι ( f lo ) ,
G out ( k , t ) = exp { 0 t ln k ( t ) d Y out ( t ) } ,
Y out ( t ) = 0 t d Λ out ( t ) + r ( t ) ε d A out ( t ) + r ( t ) ¯ ε d A out ( t ) + 1 ε 2 d t ,
[ Y out ( t ) , Y out ( t ) ] = 0     t , t 0.
d d t g ( k , t ) [ Z ] = g ( k , t ) [ ( K + μ b f ( t ) + μ b r ( t ) ¯ / ε + 1 / 2 | f ( t ) + r ( t ) / ε | 2 ) Z Z ( K + μ b f ( t ) + μ b r ( t ) ¯ / ε + 1 / 2 | f ( t ) + r ( t ) / ε | 2 ) + k ( t ) ( μ b + f ( t ) ¯ + r ( t ) ¯ / ε ) Z ( μ b + f ( t ) + r ( t ) / ε ) ] ,
g ( k , t ) [ Z ] = κ Ω t k ( κ ) V ( κ | t ) Z V ( κ | t ) d κ ,
V ( κ | t ) = e R ( t ) S ( t n ) S ( t 1 ) ,
S ( t ) = e R ( t ) ( μ b + f ( t ) + r ( t ) ε ) e R ( t ) ,
R ( t ) = K t + 0 t ( μ b f ( t ) + μ b r ( t ) ¯ / ε + 1 / 2 | f ( t ) + r ( t ) / ε | 2 ) d t .
V ^ ( t ) = e R ( t ) n = 0 0 t d t n 0 t n d t n 1 0 t 2 d t 1 S ( t n ) S ( t 1 ) i = 1 n d Y ( t i ) ,
S ( t ) = e R ( t ) ( μ b + f ( t ) + r ( t ) ε I ) e R ( t ) ,
d V ^ ( t ) = R ( t ) V ^ ( t ) d t + ( μ b + f ( t ) + r ( t ) ε I ) V ^ ( t ) d Y ( t ) , V ^ ( 0 ) = I .
d ψ ^ ( t ) = R ( t ) ψ ^ ( t ) d t + ( μ b + f ( t ) + r ( t ) ε I ) ψ ^ ( t ) d Y ( t ) , ψ ^ ( 0 ) = ψ .
d φ ^ ( t ) = ( K + μ b f ( t ) + μ b r ( t ) ¯ / ε ) φ ^ ( t ) d t + ( μ / 2 b b t + μ Re ( b t ( f ( t ) ¯ + r ( t ) ¯ / ε ) ) ) φ ^ ( t ) d t + ( ( μ b b t + 2 μ Re ( b t ( f ( t ) ¯ + r ( t ) ¯ / ε ) ) + | f ( t ) + r ( t ) / ε | 2 ) 1 / 2 × ( μ b + f ( t ) + r ( t ) / ε ) I ) φ ^ ( t ) d Y ( t ) ,
d ρ ^ ( t ) = ( i [ H , ρ ^ ( t ) ] μ 2 { b b , ρ ^ ( t ) } + μ b ρ ^ ( t ) b + μ [ b f ( t ) ¯ b f ( t ) , ρ ^ ( t ) ] ) d t + ( ε μ b ρ ^ ( t ) b + μ b ( ε f ( t ) ¯ + r ( t ) ¯ ) ρ ^ ( t ) + ρ ^ ( t ) μ b ( ε f ( t ) + r ( t ) ) ε μ b b t ρ ^ ( t ) 2 μ Re ( b t ( ε f ( t ) ¯ + r ( t ) ¯ ) ) ρ ^ ( t ) ) × ( ε d Y ( t ) ε μ b b t d t 2 μ Re ( b t ( ε f ( t ) ¯ + r ( t ) ¯ ) ) d t ε 1 | ε f ( t ) + r ( t ) | 2 d t ) × ( ε 2 μ b b t + 2 μ Re ( b t ( ε 2 f ( t ) ¯ + ε r ( t ) ¯ ) ) + | ε f ( t ) + r ( t ) | 2 ) 1 .
d Y ( t ) ( κ ) = ( μ b b t + 2 μ ε 1 Re ( b t ( ε f ( t ) ¯ + r ( t ) ¯ ) ) + ε 2 | ε f ( t ) + r ( t ) ¯ | 2 ) d t ,
d W ε ( t ) = ε d Y ( t ) d t ε ,
d W ε ( t ) d W ε ( t ) = ε 2 d Y ( t ) = ε d W ε ( t ) + d t , d W ε ( t ) d t = 0 ,
d W ( t ) d W ( t ) = d t , d W ( t ) d t = 0.
d ρ ^ ( t ) = ( i [ H , ρ ^ ( t ) ] μ 2 { b b , ρ ^ ( t ) } + μ b ρ ^ ( t ) b + μ [ b f ( t ) ¯ b f ( t ) , ρ ^ ( t ) ] ) d t + ( μ r ( t ) ¯ ( b b t ) ρ ^ ( t ) + μ r ( t ) ρ ^ ( t ) ( b b t ) ) × ( d W ( t ) 2 Re ( μ b t r ( t ) ¯ + f ( t ) r ( t ) ¯ ) d t ) .
d φ ^ ( t ) = ( K + μ b f ( t ) ¯ μ b f ( t ) + μ b t b μ 2 | b t | 2 ) φ ^ ( t ) d t + μ r ( t ) ¯ ( b b t ) φ ^ ( t ) ( d W ( t ) 2 Re ( μ b t r ( t ) ¯ + f ( t ) r ( t ) ¯ ) d t ) .

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