Abstract

We present the Belavkin filtering equation for the intense balanced heterodyne detection in a unitary model of an indirect observation. The measuring apparatus modelled by a Bose field is initially prepared in a coherent state and the observed process is a diffusion one. We prove that this filtering equation is relaxing: any initial square-integrable function tends asymptotically to a coherent state with an amplitude depending on the coupling constant and the initial state of the apparatus. The time development of a squeezed coherent state is studied and compared with the previous results obtained for the measuring apparatus prepared initially in the vacuum state.

© 2012 Optical Society of America

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  1. R. L. Hudson and K. R. Parthasarathy, “Quantum Ito’s formula and stochastic evolutions,” Commun. Math. Phys. 93, 301–323 (1984).
    [CrossRef]
  2. K. R. Parthasarathy, An Introduction to Quantum Stochastic Calculus (Birkhäuser, 1992).
  3. V. P. Belavkin, “A posterior Schrödinger equation for continuous nondemolition measurement,” J. Math. Phys. 31, 2930–2934 (1990).
    [CrossRef]
  4. A. Barchielli and V. Belavkin, “Measurement continuous in time and a posteriori states in quantum mechanics,” J. Phys. A: Math. Gen. 24, 1495–1514 (1991).
    [CrossRef]
  5. V. P. Belavkin, “Measurement, filtering and control in quantum open dynamical systems,” Rep. Math. Phys. 43, A405–A425 (1999).
    [CrossRef]
  6. V. P. Belavkin, “Quantum causality, decoherence, trajectories and information,” Rep. Prog. Phys. 65, 353–420 (2002).
    [CrossRef]
  7. A. Barchielli and A. M. Paganoni, “Detection theory in quantum optics: stochastic representation,” Quantum Opt. 8, 133–156 (1996).
    [CrossRef]
  8. J. E. Gough and C. Köstler, “Quantum filtering in coherent states,” Commun. Stoch. Anal. 4, 505–521 (2010).
  9. A. Da¸browska and P. Staszewski, “Filtering equation for measurement of a coherent channel,” J. Opt. Soc. Am. B 28, 1238–1244 (2011).
    [CrossRef]
  10. Ch. Gerry and P. Knight, Introductory Quantum Optics(Cambridge University, 2005).
  11. A. Da¸browska and P. Staszewski, “Squeezed coherent state undergoing a continuous nondemolition observation,” Phys. Lett. A 375, 3950–3955 (2011).
    [CrossRef]
  12. H. Carmichael, An Open Systems Approach to Quantum Optics (Springer-Verlag, 1993).
  13. A. C. Doherty and H. Mabuchi, “Atoms in microcavities: quantum electrodynamics, quantum statistical mechanics, and quantum information science,” in Optical Microcavities, K. Vahala, ed. (World Scientific Press, 2004), pp. 367–414.
  14. H. Wiseman and G. J. Milburn, Quantum Measurement and Control (Cambridge University, 2010).
  15. T. Brun, “A single model of quantum trajectories,” Am. J. Phys. 70, 719–737 (2002).
    [CrossRef]
  16. R. van Handel, J. K. Stockton, and H. Mabuchi, “Modelling and feedback control design for quantum state preparation,” J. Opt. B: Quantum Semiclass. Opt. 7, S179–S197 (2005).
    [CrossRef]
  17. K. Jacobs and D. Steck, “A straightforward introduction to continuous quantum measurement,” Contemp. Phys. 47, 279–303 (2006).
    [CrossRef]
  18. L. Bouten, R. van Handel, and M. James, “An introduction to quantum filtering,” SIAM J. Control Optim. 46, 2199–2241 (2007).
    [CrossRef]
  19. A. Barchielli, “Continual measurements in quantum mechanics and quantum stochastic calculus,” in Open Quantum Systems III. Recent Developments, S. Attal, A. Joye, and C.-A. Pillet, eds. (Springer, 2006), pp. 207–288.
  20. V. P. Belavkin and M. Guţă, eds. Quantum Stochastics and Information (World Scientific, 2006).
  21. A. Barchielli and M. Gregoratti, Quantum Trajectories and Measurement in Continuous Time: The Diffusive Case(Springer-Verlag, 2009).
  22. C. W. Gardiner and P. Zoller, Quantum Noise (Springer-Verlag, 2010).
  23. H. Mabuchi, J. Ye, and H. J. Kimble, “Full observation of single-atom dynamics in cavity QED,” Appl. Phys. B 68, 1095–1108 (1999).
    [CrossRef]
  24. M. A. Armen, J. K. Au, J. K. Stockton, A. C. Doherty, and H. Mabuchi, “Adaptive homodyne measurement of optical phase,” Phys. Rev. Lett. 89, 133602 (2002).
    [CrossRef]
  25. J. M. Geremia, J. K. Stockton, A. C. Doherty, and H. Mabuchi, “Quantum Kalman filtering and the Heisenberg limit in atomic magnetometry,” Phys. Rev. Lett. 91, 250801 (2003).
    [CrossRef]
  26. J. M. Geremia, J. K. Stockton, and H. Mabuchi, “Tensor polarizability and dispersive quantum measurement of multilevel atoms,” Phys. Rev. A 73, 042112 (2006).
    [CrossRef]
  27. P. Goetsch and R. Graham, “Linear stochastic wave equation for continuously measured quantum system,” Phys. Rev. A 50, 5242–5255 (1994).
    [CrossRef]
  28. M. J. Collet and C. W. Gardiner, “Input and output in damped quantum systems: quantum stochastic differential equations and the master equation,” Phys. Rev. A 31, 3761–3774 (1985).
    [CrossRef]
  29. A. Barchielli, “Direct and heterodyne detection and other applications of quantum stochastic calculus to quantum optics,” Quantum Opt. 2, 423–441 (1990).
    [CrossRef]

2011 (2)

A. Da¸browska and P. Staszewski, “Squeezed coherent state undergoing a continuous nondemolition observation,” Phys. Lett. A 375, 3950–3955 (2011).
[CrossRef]

A. Da¸browska and P. Staszewski, “Filtering equation for measurement of a coherent channel,” J. Opt. Soc. Am. B 28, 1238–1244 (2011).
[CrossRef]

2010 (1)

J. E. Gough and C. Köstler, “Quantum filtering in coherent states,” Commun. Stoch. Anal. 4, 505–521 (2010).

2007 (1)

L. Bouten, R. van Handel, and M. James, “An introduction to quantum filtering,” SIAM J. Control Optim. 46, 2199–2241 (2007).
[CrossRef]

2006 (2)

K. Jacobs and D. Steck, “A straightforward introduction to continuous quantum measurement,” Contemp. Phys. 47, 279–303 (2006).
[CrossRef]

J. M. Geremia, J. K. Stockton, and H. Mabuchi, “Tensor polarizability and dispersive quantum measurement of multilevel atoms,” Phys. Rev. A 73, 042112 (2006).
[CrossRef]

2005 (1)

R. van Handel, J. K. Stockton, and H. Mabuchi, “Modelling and feedback control design for quantum state preparation,” J. Opt. B: Quantum Semiclass. Opt. 7, S179–S197 (2005).
[CrossRef]

2003 (1)

J. M. Geremia, J. K. Stockton, A. C. Doherty, and H. Mabuchi, “Quantum Kalman filtering and the Heisenberg limit in atomic magnetometry,” Phys. Rev. Lett. 91, 250801 (2003).
[CrossRef]

2002 (3)

M. A. Armen, J. K. Au, J. K. Stockton, A. C. Doherty, and H. Mabuchi, “Adaptive homodyne measurement of optical phase,” Phys. Rev. Lett. 89, 133602 (2002).
[CrossRef]

T. Brun, “A single model of quantum trajectories,” Am. J. Phys. 70, 719–737 (2002).
[CrossRef]

V. P. Belavkin, “Quantum causality, decoherence, trajectories and information,” Rep. Prog. Phys. 65, 353–420 (2002).
[CrossRef]

1999 (2)

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

H. Mabuchi, J. Ye, and H. J. Kimble, “Full observation of single-atom dynamics in cavity QED,” Appl. Phys. B 68, 1095–1108 (1999).
[CrossRef]

1996 (1)

A. Barchielli and A. M. Paganoni, “Detection theory in quantum optics: stochastic representation,” Quantum Opt. 8, 133–156 (1996).
[CrossRef]

1994 (1)

P. Goetsch and R. Graham, “Linear stochastic wave equation for continuously measured quantum system,” Phys. Rev. A 50, 5242–5255 (1994).
[CrossRef]

1991 (1)

A. Barchielli and V. Belavkin, “Measurement 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]

1985 (1)

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

1984 (1)

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

Armen, M. A.

M. A. Armen, J. K. Au, J. K. Stockton, A. C. Doherty, and H. Mabuchi, “Adaptive homodyne measurement of optical phase,” Phys. Rev. Lett. 89, 133602 (2002).
[CrossRef]

Au, J. K.

M. A. Armen, J. K. Au, J. K. Stockton, A. C. Doherty, and H. Mabuchi, “Adaptive homodyne measurement of optical phase,” Phys. Rev. Lett. 89, 133602 (2002).
[CrossRef]

Barchielli, A.

A. Barchielli and A. M. Paganoni, “Detection theory in quantum optics: stochastic representation,” Quantum Opt. 8, 133–156 (1996).
[CrossRef]

A. Barchielli and V. Belavkin, “Measurement 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 and M. Gregoratti, Quantum Trajectories and Measurement in Continuous Time: The Diffusive Case(Springer-Verlag, 2009).

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

Belavkin, V.

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

Belavkin, V. P.

V. P. Belavkin, “Quantum causality, decoherence, trajectories and information,” Rep. Prog. Phys. 65, 353–420 (2002).
[CrossRef]

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

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

Bouten, L.

L. Bouten, R. van Handel, and M. James, “An introduction to quantum filtering,” SIAM J. Control Optim. 46, 2199–2241 (2007).
[CrossRef]

Brun, T.

T. Brun, “A single model of quantum trajectories,” Am. J. Phys. 70, 719–737 (2002).
[CrossRef]

Carmichael, H.

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

Collet, M. J.

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

Da¸browska, A.

A. Da¸browska and P. Staszewski, “Squeezed coherent state undergoing a continuous nondemolition observation,” Phys. Lett. A 375, 3950–3955 (2011).
[CrossRef]

A. Da¸browska and P. Staszewski, “Filtering equation for measurement of a coherent channel,” J. Opt. Soc. Am. B 28, 1238–1244 (2011).
[CrossRef]

Doherty, A. C.

J. M. Geremia, J. K. Stockton, A. C. Doherty, and H. Mabuchi, “Quantum Kalman filtering and the Heisenberg limit in atomic magnetometry,” Phys. Rev. Lett. 91, 250801 (2003).
[CrossRef]

M. A. Armen, J. K. Au, J. K. Stockton, A. C. Doherty, and H. Mabuchi, “Adaptive homodyne measurement of optical phase,” Phys. Rev. Lett. 89, 133602 (2002).
[CrossRef]

A. C. Doherty and H. Mabuchi, “Atoms in microcavities: quantum electrodynamics, quantum statistical mechanics, and quantum information science,” in Optical Microcavities, K. Vahala, ed. (World Scientific Press, 2004), pp. 367–414.

Gardiner, C. W.

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

C. W. Gardiner and P. Zoller, Quantum Noise (Springer-Verlag, 2010).

Geremia, J. M.

J. M. Geremia, J. K. Stockton, and H. Mabuchi, “Tensor polarizability and dispersive quantum measurement of multilevel atoms,” Phys. Rev. A 73, 042112 (2006).
[CrossRef]

J. M. Geremia, J. K. Stockton, A. C. Doherty, and H. Mabuchi, “Quantum Kalman filtering and the Heisenberg limit in atomic magnetometry,” Phys. Rev. Lett. 91, 250801 (2003).
[CrossRef]

Gerry, Ch.

Ch. Gerry and P. Knight, Introductory Quantum Optics(Cambridge University, 2005).

Goetsch, P.

P. Goetsch and R. Graham, “Linear stochastic wave equation for continuously measured quantum system,” Phys. Rev. A 50, 5242–5255 (1994).
[CrossRef]

Gough, J. E.

J. E. Gough and C. Köstler, “Quantum filtering in coherent states,” Commun. Stoch. Anal. 4, 505–521 (2010).

Graham, R.

P. Goetsch and R. Graham, “Linear stochastic wave equation for continuously measured quantum system,” Phys. Rev. A 50, 5242–5255 (1994).
[CrossRef]

Gregoratti, M.

A. Barchielli and M. Gregoratti, Quantum Trajectories and Measurement in Continuous Time: The Diffusive Case(Springer-Verlag, 2009).

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]

Jacobs, K.

K. Jacobs and D. Steck, “A straightforward introduction to continuous quantum measurement,” Contemp. Phys. 47, 279–303 (2006).
[CrossRef]

James, M.

L. Bouten, R. van Handel, and M. James, “An introduction to quantum filtering,” SIAM J. Control Optim. 46, 2199–2241 (2007).
[CrossRef]

Kimble, H. J.

H. Mabuchi, J. Ye, and H. J. Kimble, “Full observation of single-atom dynamics in cavity QED,” Appl. Phys. B 68, 1095–1108 (1999).
[CrossRef]

Knight, P.

Ch. Gerry and P. Knight, Introductory Quantum Optics(Cambridge University, 2005).

Köstler, C.

J. E. Gough and C. Köstler, “Quantum filtering in coherent states,” Commun. Stoch. Anal. 4, 505–521 (2010).

Mabuchi, H.

J. M. Geremia, J. K. Stockton, and H. Mabuchi, “Tensor polarizability and dispersive quantum measurement of multilevel atoms,” Phys. Rev. A 73, 042112 (2006).
[CrossRef]

R. van Handel, J. K. Stockton, and H. Mabuchi, “Modelling and feedback control design for quantum state preparation,” J. Opt. B: Quantum Semiclass. Opt. 7, S179–S197 (2005).
[CrossRef]

J. M. Geremia, J. K. Stockton, A. C. Doherty, and H. Mabuchi, “Quantum Kalman filtering and the Heisenberg limit in atomic magnetometry,” Phys. Rev. Lett. 91, 250801 (2003).
[CrossRef]

M. A. Armen, J. K. Au, J. K. Stockton, A. C. Doherty, and H. Mabuchi, “Adaptive homodyne measurement of optical phase,” Phys. Rev. Lett. 89, 133602 (2002).
[CrossRef]

H. Mabuchi, J. Ye, and H. J. Kimble, “Full observation of single-atom dynamics in cavity QED,” Appl. Phys. B 68, 1095–1108 (1999).
[CrossRef]

A. C. Doherty and H. Mabuchi, “Atoms in microcavities: quantum electrodynamics, quantum statistical mechanics, and quantum information science,” in Optical Microcavities, K. Vahala, ed. (World Scientific Press, 2004), pp. 367–414.

Milburn, G. J.

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

Paganoni, A. M.

A. Barchielli and A. M. Paganoni, “Detection theory in quantum optics: stochastic representation,” Quantum Opt. 8, 133–156 (1996).
[CrossRef]

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, 1992).

Staszewski, P.

A. Da¸browska and P. Staszewski, “Filtering equation for measurement of a coherent channel,” J. Opt. Soc. Am. B 28, 1238–1244 (2011).
[CrossRef]

A. Da¸browska and P. Staszewski, “Squeezed coherent state undergoing a continuous nondemolition observation,” Phys. Lett. A 375, 3950–3955 (2011).
[CrossRef]

Steck, D.

K. Jacobs and D. Steck, “A straightforward introduction to continuous quantum measurement,” Contemp. Phys. 47, 279–303 (2006).
[CrossRef]

Stockton, J. K.

J. M. Geremia, J. K. Stockton, and H. Mabuchi, “Tensor polarizability and dispersive quantum measurement of multilevel atoms,” Phys. Rev. A 73, 042112 (2006).
[CrossRef]

R. van Handel, J. K. Stockton, and H. Mabuchi, “Modelling and feedback control design for quantum state preparation,” J. Opt. B: Quantum Semiclass. Opt. 7, S179–S197 (2005).
[CrossRef]

J. M. Geremia, J. K. Stockton, A. C. Doherty, and H. Mabuchi, “Quantum Kalman filtering and the Heisenberg limit in atomic magnetometry,” Phys. Rev. Lett. 91, 250801 (2003).
[CrossRef]

M. A. Armen, J. K. Au, J. K. Stockton, A. C. Doherty, and H. Mabuchi, “Adaptive homodyne measurement of optical phase,” Phys. Rev. Lett. 89, 133602 (2002).
[CrossRef]

van Handel, R.

L. Bouten, R. van Handel, and M. James, “An introduction to quantum filtering,” SIAM J. Control Optim. 46, 2199–2241 (2007).
[CrossRef]

R. van Handel, J. K. Stockton, and H. Mabuchi, “Modelling and feedback control design for quantum state preparation,” J. Opt. B: Quantum Semiclass. Opt. 7, S179–S197 (2005).
[CrossRef]

Wiseman, H.

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

Ye, J.

H. Mabuchi, J. Ye, and H. J. Kimble, “Full observation of single-atom dynamics in cavity QED,” Appl. Phys. B 68, 1095–1108 (1999).
[CrossRef]

Zoller, P.

C. W. Gardiner and P. Zoller, Quantum Noise (Springer-Verlag, 2010).

Am. J. Phys. (1)

T. Brun, “A single model of quantum trajectories,” Am. J. Phys. 70, 719–737 (2002).
[CrossRef]

Appl. Phys. B (1)

H. Mabuchi, J. Ye, and H. J. Kimble, “Full observation of single-atom dynamics in cavity QED,” Appl. Phys. B 68, 1095–1108 (1999).
[CrossRef]

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]

Commun. Stoch. Anal. (1)

J. E. Gough and C. Köstler, “Quantum filtering in coherent states,” Commun. Stoch. Anal. 4, 505–521 (2010).

Contemp. Phys. (1)

K. Jacobs and D. Steck, “A straightforward introduction to continuous quantum measurement,” Contemp. Phys. 47, 279–303 (2006).
[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. Opt. B: Quantum Semiclass. Opt. (1)

R. van Handel, J. K. Stockton, and H. Mabuchi, “Modelling and feedback control design for quantum state preparation,” J. Opt. B: Quantum Semiclass. Opt. 7, S179–S197 (2005).
[CrossRef]

J. Opt. Soc. Am. B (1)

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

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

Phys. Lett. A (1)

A. Da¸browska and P. Staszewski, “Squeezed coherent state undergoing a continuous nondemolition observation,” Phys. Lett. A 375, 3950–3955 (2011).
[CrossRef]

Phys. Rev. A (3)

J. M. Geremia, J. K. Stockton, and H. Mabuchi, “Tensor polarizability and dispersive quantum measurement of multilevel atoms,” Phys. Rev. A 73, 042112 (2006).
[CrossRef]

P. Goetsch and R. Graham, “Linear stochastic wave equation for continuously measured quantum system,” Phys. Rev. A 50, 5242–5255 (1994).
[CrossRef]

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

Phys. Rev. Lett. (2)

M. A. Armen, J. K. Au, J. K. Stockton, A. C. Doherty, and H. Mabuchi, “Adaptive homodyne measurement of optical phase,” Phys. Rev. Lett. 89, 133602 (2002).
[CrossRef]

J. M. Geremia, J. K. Stockton, A. C. Doherty, and H. Mabuchi, “Quantum Kalman filtering and the Heisenberg limit in atomic magnetometry,” Phys. Rev. Lett. 91, 250801 (2003).
[CrossRef]

Quantum Opt. (2)

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 and A. M. Paganoni, “Detection theory in quantum optics: stochastic representation,” Quantum Opt. 8, 133–156 (1996).
[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]

Rep. Prog. Phys. (1)

V. P. Belavkin, “Quantum causality, decoherence, trajectories and information,” Rep. Prog. Phys. 65, 353–420 (2002).
[CrossRef]

SIAM J. Control Optim. (1)

L. Bouten, R. van Handel, and M. James, “An introduction to quantum filtering,” SIAM J. Control Optim. 46, 2199–2241 (2007).
[CrossRef]

Other (9)

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

V. P. Belavkin and M. Guţă, eds. Quantum Stochastics and Information (World Scientific, 2006).

A. Barchielli and M. Gregoratti, Quantum Trajectories and Measurement in Continuous Time: The Diffusive Case(Springer-Verlag, 2009).

C. W. Gardiner and P. Zoller, Quantum Noise (Springer-Verlag, 2010).

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

A. C. Doherty and H. Mabuchi, “Atoms in microcavities: quantum electrodynamics, quantum statistical mechanics, and quantum information science,” in Optical Microcavities, K. Vahala, ed. (World Scientific Press, 2004), pp. 367–414.

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

Ch. Gerry and P. Knight, Introductory Quantum Optics(Cambridge University, 2005).

K. R. Parthasarathy, An Introduction to Quantum Stochastic Calculus (Birkhäuser, 1992).

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