Abstract

Depolarization of quantum fields is handled through a master equation of the Lindblad type. The specific feature of the model is that it couples the field modes to a randomly distributed atomic bath dispersively. The depolarizing dynamics emerging from this approach is analyzed for relevant states.

© 2006 Optical Society of America

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  1. A. Aspect, P. Grangier, and G. Roger, "Experimental tests of realistic local theories via Bell's theorem," Phys. Rev. Lett. 47, 460-463 (1981).
    [CrossRef]
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    [CrossRef] [PubMed]
  3. T. Tsegaye, J. Söderholm, M. Atatüre, A. Trifonov, G. Björk, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, "Experimental demonstration of three mutually orthogonal polarization states of entangled photons," Phys. Rev. Lett. 85, 5013-5017 (2000).
    [CrossRef] [PubMed]
  4. M. Barbieri, F. De Martini, G. Di Nepi, P. Mataloni, G. M. D'Ariano, and C. Macchiavello, "Detection of entanglement with polarized photons: experimental realization of an entanglement witness," Phys. Rev. Lett. 91, 227901 (2003).
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  7. D. N. Klyshko, "Multiphoton interference and polarization effects," Phys. Lett. A 163, 349-355 (1992).
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  8. P. Usachev, J. Söderholm, G. Björk, and A. Trifonov, "Experimental verification of differences between classical and quantum polarization properties," Opt. Commun. 193, 161-173 (2001).
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  9. M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).
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    [CrossRef] [PubMed]
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  16. A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and W. Zwerger, "Dynamics of the dissipative two-state system," Rev. Mod. Phys. 59, 1-85 (1987).
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  21. A. V. Uskov, A. P. Jauho, B. Tromborg, J. Mørk, and R. Lang, "Dephasing times in quantum dots due to elastic LO phonon-carrier collisions," Phys. Rev. Lett. 85, 1516-1519 (2000).
    [CrossRef] [PubMed]
  22. B. Krummheuer, V. M. Axt, and T. Kuhn, "Theory of pure dephasing and the resulting absorption line shape in semiconductor quantum dots," Phys. Rev. B 65, 195313 (2002).
    [CrossRef]
  23. E. Pazy, "Calculation of pure dephasing for excitons in quantum dots," Semicond. Sci. Technol. 17, 1172-1179 (2002).
    [CrossRef]
  24. Y. Makhlin, G. Schön, and A. Shnirman, "Josephson-junction qubits with controlled couplings," Nature 398, 305-307 (1999).
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  25. Y. Nakamura, Yu. A. Pashkin, and J. S. Tsai, "Coherent control of macroscopic quantum states in a single-Cooper-pair box," Nature 398, 786-788 (1999).
    [CrossRef]
  26. C. H. van der Wal, A. C. J. ter Haar, F. K. Wilhelm, R. N. Schouten, C. J. P. M. Harmans, T. P. Orlando, S. Lloyd, and J. E. Mooij, "Quantum superposition of macroscopic persistent-current states," Science 290, 773-777 (2000).
    [CrossRef] [PubMed]
  27. J. H. Reina, L. Quiroga, and N. F. Johnson, "Decoherence of quantum registers," Phys. Rev. A 65, 032326 (2002).
    [CrossRef]
  28. V. P. Karassiov, "Polarization structure of quantum light fields: a new insight. I. General outlook," J. Phys. A 26, 4345-4354 (1993).
    [CrossRef]
  29. M. G. Raymer and A. C. Funk, "Quantum-state tomography of two-mode light using generalized rotations in phase space," Phys. Rev. A 61, 015801 (1999).
    [CrossRef]
  30. To avoid any misunderstanding, we note that what we are calling here single-mode fields are also known as two-mode fields, especially when dealing with polarization properties.
  31. H. Prakash and N. Chandra, "Density operator of unpolarized radiation," Phys. Rev. A 4, 796-799 (1971).
    [CrossRef]
  32. G. S. Agarwal, "On the state of unpolarized radiation," Lett. Nuovo Cimento 1, 53-56 (1971).
    [CrossRef]
  33. J. Lehner, U. Leonhardt, and H. Paul, "Unpolarized light: classical and quantum states," Phys. Rev. A 53, 2727-2735 (1996).
    [CrossRef] [PubMed]
  34. J. Söderholm, G. Björk, and A. Trifonov, "Unpolarized light in quantum optics," Opt. Spectrosc. 91, 532-534 (2001).
    [CrossRef]
  35. A. Wünsche, "Quantum-mechanical description of polarized and unpolarized light," Fortschr. Phys. 51, 262-266 (2003).
    [CrossRef]
  36. G. Lindblad, "On the generators of quantum dynamical semigroups," Commun. Math. Phys. 48, 119-130 (1976).
    [CrossRef]
  37. K. Kraus, "General state changes in quantum theory," Ann. Phys. (N.Y.) 64, 311-335 (1971).
    [CrossRef]
  38. V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, "Completely positive dynamical semigroups of N-level systems," J. Math. Phys. 17, 821-825 (1976).
    [CrossRef]
  39. M. Orszag, Quantum Optics (Springer, 2000).
  40. A. B. Klimov, L. L. Sánchez-Soto, A. Navarro, and E. C. Yustas, "Effective Hamiltonians in quantum optics: a systematic approach," J. Mod. Opt. 49, 2211-2226 (2002).
    [CrossRef]
  41. H. P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford U. Press, 2002).
  42. A. G. Kofman, "Absorption spectrum of an atom strongly coupled to a high-temperature reservoir," Phys. Rev. A 71, 033806 (2005).
    [CrossRef]
  43. M. Jakob and S. Stenholm, "Effects of a thermal reservoir on variational functions in open systems," Phys. Rev. A 70, 012104 (2004).
    [CrossRef]
  44. A. Galindo and M. A. Martín-Delgado, "Information and computation: classical and quantum aspects," Rev. Mod. Phys. 74, 347-423 (2002).
    [CrossRef]
  45. A. B. Klimov and L. L. Sánchez-Soto, "Method of small rotations and effective Hamiltonians in nonlinear quantum optics," Phys. Rev. A 61, 063802 (2000).
    [CrossRef]
  46. A. B. Klimov, J. L. Romero, and C. Saavedra, "General properties of quantum systems interacting with a field mode in a low-Q cavity," Phys. Rev. A 64, 063802 (2001).
    [CrossRef]
  47. A. B. Klimov, J. L. Romero, J. Delgado, and L. L. Sánchez-Soto, "Master equations for effective Hamiltonians," J. Opt. B 5, 34-39 (2003).
    [CrossRef]

2005 (2)

A. B. Klimov, L. L. Sánchez-Soto, E. C. Yustas, J. Söderholm, and G. Björk, "Distance-based degrees of polarization for a quantum field," Phys. Rev. A 72, 033813 (2005).
[CrossRef]

A. G. Kofman, "Absorption spectrum of an atom strongly coupled to a high-temperature reservoir," Phys. Rev. A 71, 033806 (2005).
[CrossRef]

2004 (3)

M. Jakob and S. Stenholm, "Effects of a thermal reservoir on variational functions in open systems," Phys. Rev. A 70, 012104 (2004).
[CrossRef]

S. G. Schirmer and A. I. Solomon, "Constraints on relaxation rates for N-level quantum systems," Phys. Rev. A 70, 022107 (2004).
[CrossRef]

G. A. Durkin, C. Simon, J. Eisert, and D. Bouwmeester, "Resilience of multiphoton entanglement under losses," Phys. Rev. A 70, 062305 (2004).
[CrossRef]

2003 (4)

M. Barbieri, F. De Martini, G. Di Nepi, P. Mataloni, G. M. D'Ariano, and C. Macchiavello, "Detection of entanglement with polarized photons: experimental realization of an entanglement witness," Phys. Rev. Lett. 91, 227901 (2003).
[CrossRef] [PubMed]

M. Legré, M. Wegmüller, and N. Gisin, "Quantum measurement of the degree of polarization of a light beam," Phys. Rev. Lett. 91, 167902 (2003).
[CrossRef] [PubMed]

A. Wünsche, "Quantum-mechanical description of polarized and unpolarized light," Fortschr. Phys. 51, 262-266 (2003).
[CrossRef]

A. B. Klimov, J. L. Romero, J. Delgado, and L. L. Sánchez-Soto, "Master equations for effective Hamiltonians," J. Opt. B 5, 34-39 (2003).
[CrossRef]

2002 (6)

A. B. Klimov, L. L. Sánchez-Soto, A. Navarro, and E. C. Yustas, "Effective Hamiltonians in quantum optics: a systematic approach," J. Mod. Opt. 49, 2211-2226 (2002).
[CrossRef]

A. Galindo and M. A. Martín-Delgado, "Information and computation: classical and quantum aspects," Rev. Mod. Phys. 74, 347-423 (2002).
[CrossRef]

J. H. Reina, L. Quiroga, and N. F. Johnson, "Decoherence of quantum registers," Phys. Rev. A 65, 032326 (2002).
[CrossRef]

T. Yu and J. H. Eberly, "Phonon decoherence of quantum entanglement: robust and fragile states," Phys. Rev. B 66, 193306 (2002).
[CrossRef]

B. Krummheuer, V. M. Axt, and T. Kuhn, "Theory of pure dephasing and the resulting absorption line shape in semiconductor quantum dots," Phys. Rev. B 65, 195313 (2002).
[CrossRef]

E. Pazy, "Calculation of pure dephasing for excitons in quantum dots," Semicond. Sci. Technol. 17, 1172-1179 (2002).
[CrossRef]

2001 (3)

P. Usachev, J. Söderholm, G. Björk, and A. Trifonov, "Experimental verification of differences between classical and quantum polarization properties," Opt. Commun. 193, 161-173 (2001).
[CrossRef]

J. Söderholm, G. Björk, and A. Trifonov, "Unpolarized light in quantum optics," Opt. Spectrosc. 91, 532-534 (2001).
[CrossRef]

A. B. Klimov, J. L. Romero, and C. Saavedra, "General properties of quantum systems interacting with a field mode in a low-Q cavity," Phys. Rev. A 64, 063802 (2001).
[CrossRef]

2000 (5)

A. B. Klimov and L. L. Sánchez-Soto, "Method of small rotations and effective Hamiltonians in nonlinear quantum optics," Phys. Rev. A 61, 063802 (2000).
[CrossRef]

A. V. Uskov, A. P. Jauho, B. Tromborg, J. Mørk, and R. Lang, "Dephasing times in quantum dots due to elastic LO phonon-carrier collisions," Phys. Rev. Lett. 85, 1516-1519 (2000).
[CrossRef] [PubMed]

C. H. van der Wal, A. C. J. ter Haar, F. K. Wilhelm, R. N. Schouten, C. J. P. M. Harmans, T. P. Orlando, S. Lloyd, and J. E. Mooij, "Quantum superposition of macroscopic persistent-current states," Science 290, 773-777 (2000).
[CrossRef] [PubMed]

T. Tsegaye, J. Söderholm, M. Atatüre, A. Trifonov, G. Björk, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, "Experimental demonstration of three mutually orthogonal polarization states of entangled photons," Phys. Rev. Lett. 85, 5013-5017 (2000).
[CrossRef] [PubMed]

A. Luis and L. L. Sánchez-Soto, "Quantum phase difference, phase measurements and quantum Stokes parameters," Prog. Opt. 41, 421-481 (2000).
[CrossRef]

1999 (5)

A. P. Alodjants and S. M. Arakelian, "Quantum phase measurements and non-classical polarization states of light," J. Mod. Opt. 46, 475-507 (1999).

Y. Makhlin, G. Schön, and A. Shnirman, "Josephson-junction qubits with controlled couplings," Nature 398, 305-307 (1999).
[CrossRef]

Y. Nakamura, Yu. A. Pashkin, and J. S. Tsai, "Coherent control of macroscopic quantum states in a single-Cooper-pair box," Nature 398, 786-788 (1999).
[CrossRef]

T. Takagahara, "Theory of exciton dephasing in semiconductor quantum dots," Phys. Rev. B 60, 2638-2652 (1999).
[CrossRef]

M. G. Raymer and A. C. Funk, "Quantum-state tomography of two-mode light using generalized rotations in phase space," Phys. Rev. A 61, 015801 (1999).
[CrossRef]

1996 (1)

J. Lehner, U. Leonhardt, and H. Paul, "Unpolarized light: classical and quantum states," Phys. Rev. A 53, 2727-2735 (1996).
[CrossRef] [PubMed]

1995 (1)

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, "New high-intensity source of polarization-entangled photon pairs," Phys. Rev. Lett. 75, 4337-4341 (1995).
[CrossRef] [PubMed]

1993 (1)

V. P. Karassiov, "Polarization structure of quantum light fields: a new insight. I. General outlook," J. Phys. A 26, 4345-4354 (1993).
[CrossRef]

1992 (1)

D. N. Klyshko, "Multiphoton interference and polarization effects," Phys. Lett. A 163, 349-355 (1992).
[CrossRef]

1987 (1)

A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and W. Zwerger, "Dynamics of the dissipative two-state system," Rev. Mod. Phys. 59, 1-85 (1987).
[CrossRef]

1981 (2)

A. O. Caldeira and A. J. Leggett, "Influence of dissipation on quantum tunneling in macroscopic systems," Phys. Rev. Lett. 46, 211-214 (1981).
[CrossRef]

A. Aspect, P. Grangier, and G. Roger, "Experimental tests of realistic local theories via Bell's theorem," Phys. Rev. Lett. 47, 460-463 (1981).
[CrossRef]

1976 (2)

G. Lindblad, "On the generators of quantum dynamical semigroups," Commun. Math. Phys. 48, 119-130 (1976).
[CrossRef]

V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, "Completely positive dynamical semigroups of N-level systems," J. Math. Phys. 17, 821-825 (1976).
[CrossRef]

1971 (3)

K. Kraus, "General state changes in quantum theory," Ann. Phys. (N.Y.) 64, 311-335 (1971).
[CrossRef]

H. Prakash and N. Chandra, "Density operator of unpolarized radiation," Phys. Rev. A 4, 796-799 (1971).
[CrossRef]

G. S. Agarwal, "On the state of unpolarized radiation," Lett. Nuovo Cimento 1, 53-56 (1971).
[CrossRef]

Agarwal, G. S.

G. S. Agarwal, "On the state of unpolarized radiation," Lett. Nuovo Cimento 1, 53-56 (1971).
[CrossRef]

Alodjants, A. P.

A. P. Alodjants and S. M. Arakelian, "Quantum phase measurements and non-classical polarization states of light," J. Mod. Opt. 46, 475-507 (1999).

Arakelian, S. M.

A. P. Alodjants and S. M. Arakelian, "Quantum phase measurements and non-classical polarization states of light," J. Mod. Opt. 46, 475-507 (1999).

Aspect, A.

A. Aspect, P. Grangier, and G. Roger, "Experimental tests of realistic local theories via Bell's theorem," Phys. Rev. Lett. 47, 460-463 (1981).
[CrossRef]

Atatüre, M.

T. Tsegaye, J. Söderholm, M. Atatüre, A. Trifonov, G. Björk, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, "Experimental demonstration of three mutually orthogonal polarization states of entangled photons," Phys. Rev. Lett. 85, 5013-5017 (2000).
[CrossRef] [PubMed]

Axt, V. M.

B. Krummheuer, V. M. Axt, and T. Kuhn, "Theory of pure dephasing and the resulting absorption line shape in semiconductor quantum dots," Phys. Rev. B 65, 195313 (2002).
[CrossRef]

Barbieri, M.

M. Barbieri, F. De Martini, G. Di Nepi, P. Mataloni, G. M. D'Ariano, and C. Macchiavello, "Detection of entanglement with polarized photons: experimental realization of an entanglement witness," Phys. Rev. Lett. 91, 227901 (2003).
[CrossRef] [PubMed]

Björk, G.

A. B. Klimov, L. L. Sánchez-Soto, E. C. Yustas, J. Söderholm, and G. Björk, "Distance-based degrees of polarization for a quantum field," Phys. Rev. A 72, 033813 (2005).
[CrossRef]

P. Usachev, J. Söderholm, G. Björk, and A. Trifonov, "Experimental verification of differences between classical and quantum polarization properties," Opt. Commun. 193, 161-173 (2001).
[CrossRef]

J. Söderholm, G. Björk, and A. Trifonov, "Unpolarized light in quantum optics," Opt. Spectrosc. 91, 532-534 (2001).
[CrossRef]

T. Tsegaye, J. Söderholm, M. Atatüre, A. Trifonov, G. Björk, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, "Experimental demonstration of three mutually orthogonal polarization states of entangled photons," Phys. Rev. Lett. 85, 5013-5017 (2000).
[CrossRef] [PubMed]

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

Bouwmeester, D.

G. A. Durkin, C. Simon, J. Eisert, and D. Bouwmeester, "Resilience of multiphoton entanglement under losses," Phys. Rev. A 70, 062305 (2004).
[CrossRef]

Breuer, H. P.

H. P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford U. Press, 2002).

Brosseau, C.

C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1998).

Caldeira, A. O.

A. O. Caldeira and A. J. Leggett, "Influence of dissipation on quantum tunneling in macroscopic systems," Phys. Rev. Lett. 46, 211-214 (1981).
[CrossRef]

Chakravarty, S.

A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and W. Zwerger, "Dynamics of the dissipative two-state system," Rev. Mod. Phys. 59, 1-85 (1987).
[CrossRef]

Chandra, N.

H. Prakash and N. Chandra, "Density operator of unpolarized radiation," Phys. Rev. A 4, 796-799 (1971).
[CrossRef]

D'Ariano, G. M.

M. Barbieri, F. De Martini, G. Di Nepi, P. Mataloni, G. M. D'Ariano, and C. Macchiavello, "Detection of entanglement with polarized photons: experimental realization of an entanglement witness," Phys. Rev. Lett. 91, 227901 (2003).
[CrossRef] [PubMed]

De Martini, F.

M. Barbieri, F. De Martini, G. Di Nepi, P. Mataloni, G. M. D'Ariano, and C. Macchiavello, "Detection of entanglement with polarized photons: experimental realization of an entanglement witness," Phys. Rev. Lett. 91, 227901 (2003).
[CrossRef] [PubMed]

Delgado, J.

A. B. Klimov, J. L. Romero, J. Delgado, and L. L. Sánchez-Soto, "Master equations for effective Hamiltonians," J. Opt. B 5, 34-39 (2003).
[CrossRef]

Di Nepi, G.

M. Barbieri, F. De Martini, G. Di Nepi, P. Mataloni, G. M. D'Ariano, and C. Macchiavello, "Detection of entanglement with polarized photons: experimental realization of an entanglement witness," Phys. Rev. Lett. 91, 227901 (2003).
[CrossRef] [PubMed]

Dorsey, A. T.

A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and W. Zwerger, "Dynamics of the dissipative two-state system," Rev. Mod. Phys. 59, 1-85 (1987).
[CrossRef]

Durkin, G. A.

G. A. Durkin, C. Simon, J. Eisert, and D. Bouwmeester, "Resilience of multiphoton entanglement under losses," Phys. Rev. A 70, 062305 (2004).
[CrossRef]

Eberly, J. H.

T. Yu and J. H. Eberly, "Phonon decoherence of quantum entanglement: robust and fragile states," Phys. Rev. B 66, 193306 (2002).
[CrossRef]

Eisert, J.

G. A. Durkin, C. Simon, J. Eisert, and D. Bouwmeester, "Resilience of multiphoton entanglement under losses," Phys. Rev. A 70, 062305 (2004).
[CrossRef]

Fisher, M. P.

A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and W. Zwerger, "Dynamics of the dissipative two-state system," Rev. Mod. Phys. 59, 1-85 (1987).
[CrossRef]

Funk, A. C.

M. G. Raymer and A. C. Funk, "Quantum-state tomography of two-mode light using generalized rotations in phase space," Phys. Rev. A 61, 015801 (1999).
[CrossRef]

Galindo, A.

A. Galindo and M. A. Martín-Delgado, "Information and computation: classical and quantum aspects," Rev. Mod. Phys. 74, 347-423 (2002).
[CrossRef]

Gardiner, C. W.

C. W. Gardiner and P. Zoller, Quantum Noise (Springer, 2000).

Garg, A.

A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and W. Zwerger, "Dynamics of the dissipative two-state system," Rev. Mod. Phys. 59, 1-85 (1987).
[CrossRef]

Gisin, N.

M. Legré, M. Wegmüller, and N. Gisin, "Quantum measurement of the degree of polarization of a light beam," Phys. Rev. Lett. 91, 167902 (2003).
[CrossRef] [PubMed]

Gorini, V.

V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, "Completely positive dynamical semigroups of N-level systems," J. Math. Phys. 17, 821-825 (1976).
[CrossRef]

Grangier, P.

A. Aspect, P. Grangier, and G. Roger, "Experimental tests of realistic local theories via Bell's theorem," Phys. Rev. Lett. 47, 460-463 (1981).
[CrossRef]

Harmans, C. J.

C. H. van der Wal, A. C. J. ter Haar, F. K. Wilhelm, R. N. Schouten, C. J. P. M. Harmans, T. P. Orlando, S. Lloyd, and J. E. Mooij, "Quantum superposition of macroscopic persistent-current states," Science 290, 773-777 (2000).
[CrossRef] [PubMed]

Jakob, M.

M. Jakob and S. Stenholm, "Effects of a thermal reservoir on variational functions in open systems," Phys. Rev. A 70, 012104 (2004).
[CrossRef]

Jauch, J. M.

J. M. Jauch and F. Rohrlich, The Theory of Photons and Electrons (Addison-Wesley, 1959).

Jauho, A. P.

A. V. Uskov, A. P. Jauho, B. Tromborg, J. Mørk, and R. Lang, "Dephasing times in quantum dots due to elastic LO phonon-carrier collisions," Phys. Rev. Lett. 85, 1516-1519 (2000).
[CrossRef] [PubMed]

Johnson, N. F.

J. H. Reina, L. Quiroga, and N. F. Johnson, "Decoherence of quantum registers," Phys. Rev. A 65, 032326 (2002).
[CrossRef]

Karassiov, V. P.

V. P. Karassiov, "Polarization structure of quantum light fields: a new insight. I. General outlook," J. Phys. A 26, 4345-4354 (1993).
[CrossRef]

Klimov, A. B.

A. B. Klimov, L. L. Sánchez-Soto, E. C. Yustas, J. Söderholm, and G. Björk, "Distance-based degrees of polarization for a quantum field," Phys. Rev. A 72, 033813 (2005).
[CrossRef]

A. B. Klimov, J. L. Romero, J. Delgado, and L. L. Sánchez-Soto, "Master equations for effective Hamiltonians," J. Opt. B 5, 34-39 (2003).
[CrossRef]

A. B. Klimov, L. L. Sánchez-Soto, A. Navarro, and E. C. Yustas, "Effective Hamiltonians in quantum optics: a systematic approach," J. Mod. Opt. 49, 2211-2226 (2002).
[CrossRef]

A. B. Klimov, J. L. Romero, and C. Saavedra, "General properties of quantum systems interacting with a field mode in a low-Q cavity," Phys. Rev. A 64, 063802 (2001).
[CrossRef]

A. B. Klimov and L. L. Sánchez-Soto, "Method of small rotations and effective Hamiltonians in nonlinear quantum optics," Phys. Rev. A 61, 063802 (2000).
[CrossRef]

Klyshko, D. N.

D. N. Klyshko, "Multiphoton interference and polarization effects," Phys. Lett. A 163, 349-355 (1992).
[CrossRef]

Kofman, A. G.

A. G. Kofman, "Absorption spectrum of an atom strongly coupled to a high-temperature reservoir," Phys. Rev. A 71, 033806 (2005).
[CrossRef]

Kossakowski, A.

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C. H. van der Wal, A. C. J. ter Haar, F. K. Wilhelm, R. N. Schouten, C. J. P. M. Harmans, T. P. Orlando, S. Lloyd, and J. E. Mooij, "Quantum superposition of macroscopic persistent-current states," Science 290, 773-777 (2000).
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M. Barbieri, F. De Martini, G. Di Nepi, P. Mataloni, G. M. D'Ariano, and C. Macchiavello, "Detection of entanglement with polarized photons: experimental realization of an entanglement witness," Phys. Rev. Lett. 91, 227901 (2003).
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A. V. Uskov, A. P. Jauho, B. Tromborg, J. Mørk, and R. Lang, "Dephasing times in quantum dots due to elastic LO phonon-carrier collisions," Phys. Rev. Lett. 85, 1516-1519 (2000).
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C. H. van der Wal, A. C. J. ter Haar, F. K. Wilhelm, R. N. Schouten, C. J. P. M. Harmans, T. P. Orlando, S. Lloyd, and J. E. Mooij, "Quantum superposition of macroscopic persistent-current states," Science 290, 773-777 (2000).
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Y. Nakamura, Yu. A. Pashkin, and J. S. Tsai, "Coherent control of macroscopic quantum states in a single-Cooper-pair box," Nature 398, 786-788 (1999).
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J. Lehner, U. Leonhardt, and H. Paul, "Unpolarized light: classical and quantum states," Phys. Rev. A 53, 2727-2735 (1996).
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A. B. Klimov, L. L. Sánchez-Soto, A. Navarro, and E. C. Yustas, "Effective Hamiltonians in quantum optics: a systematic approach," J. Mod. Opt. 49, 2211-2226 (2002).
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A. B. Klimov and L. L. Sánchez-Soto, "Method of small rotations and effective Hamiltonians in nonlinear quantum optics," Phys. Rev. A 61, 063802 (2000).
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S. G. Schirmer and A. I. Solomon, "Constraints on relaxation rates for N-level quantum systems," Phys. Rev. A 70, 022107 (2004).
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Y. Makhlin, G. Schön, and A. Shnirman, "Josephson-junction qubits with controlled couplings," Nature 398, 305-307 (1999).
[CrossRef]

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C. H. van der Wal, A. C. J. ter Haar, F. K. Wilhelm, R. N. Schouten, C. J. P. M. Harmans, T. P. Orlando, S. Lloyd, and J. E. Mooij, "Quantum superposition of macroscopic persistent-current states," Science 290, 773-777 (2000).
[CrossRef] [PubMed]

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T. Tsegaye, J. Söderholm, M. Atatüre, A. Trifonov, G. Björk, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, "Experimental demonstration of three mutually orthogonal polarization states of entangled photons," Phys. Rev. Lett. 85, 5013-5017 (2000).
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A. B. Klimov, L. L. Sánchez-Soto, E. C. Yustas, J. Söderholm, and G. Björk, "Distance-based degrees of polarization for a quantum field," Phys. Rev. A 72, 033813 (2005).
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P. Usachev, J. Söderholm, G. Björk, and A. Trifonov, "Experimental verification of differences between classical and quantum polarization properties," Opt. Commun. 193, 161-173 (2001).
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J. Söderholm, G. Björk, and A. Trifonov, "Unpolarized light in quantum optics," Opt. Spectrosc. 91, 532-534 (2001).
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T. Tsegaye, J. Söderholm, M. Atatüre, A. Trifonov, G. Björk, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, "Experimental demonstration of three mutually orthogonal polarization states of entangled photons," Phys. Rev. Lett. 85, 5013-5017 (2000).
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V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, "Completely positive dynamical semigroups of N-level systems," J. Math. Phys. 17, 821-825 (1976).
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T. Tsegaye, J. Söderholm, M. Atatüre, A. Trifonov, G. Björk, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, "Experimental demonstration of three mutually orthogonal polarization states of entangled photons," Phys. Rev. Lett. 85, 5013-5017 (2000).
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C. H. van der Wal, A. C. J. ter Haar, F. K. Wilhelm, R. N. Schouten, C. J. P. M. Harmans, T. P. Orlando, S. Lloyd, and J. E. Mooij, "Quantum superposition of macroscopic persistent-current states," Science 290, 773-777 (2000).
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J. Söderholm, G. Björk, and A. Trifonov, "Unpolarized light in quantum optics," Opt. Spectrosc. 91, 532-534 (2001).
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P. Usachev, J. Söderholm, G. Björk, and A. Trifonov, "Experimental verification of differences between classical and quantum polarization properties," Opt. Commun. 193, 161-173 (2001).
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T. Tsegaye, J. Söderholm, M. Atatüre, A. Trifonov, G. Björk, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, "Experimental demonstration of three mutually orthogonal polarization states of entangled photons," Phys. Rev. Lett. 85, 5013-5017 (2000).
[CrossRef] [PubMed]

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A. V. Uskov, A. P. Jauho, B. Tromborg, J. Mørk, and R. Lang, "Dephasing times in quantum dots due to elastic LO phonon-carrier collisions," Phys. Rev. Lett. 85, 1516-1519 (2000).
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Y. Nakamura, Yu. A. Pashkin, and J. S. Tsai, "Coherent control of macroscopic quantum states in a single-Cooper-pair box," Nature 398, 786-788 (1999).
[CrossRef]

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T. Tsegaye, J. Söderholm, M. Atatüre, A. Trifonov, G. Björk, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, "Experimental demonstration of three mutually orthogonal polarization states of entangled photons," Phys. Rev. Lett. 85, 5013-5017 (2000).
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P. Usachev, J. Söderholm, G. Björk, and A. Trifonov, "Experimental verification of differences between classical and quantum polarization properties," Opt. Commun. 193, 161-173 (2001).
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A. V. Uskov, A. P. Jauho, B. Tromborg, J. Mørk, and R. Lang, "Dephasing times in quantum dots due to elastic LO phonon-carrier collisions," Phys. Rev. Lett. 85, 1516-1519 (2000).
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C. H. van der Wal, A. C. J. ter Haar, F. K. Wilhelm, R. N. Schouten, C. J. P. M. Harmans, T. P. Orlando, S. Lloyd, and J. E. Mooij, "Quantum superposition of macroscopic persistent-current states," Science 290, 773-777 (2000).
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M. Legré, M. Wegmüller, and N. Gisin, "Quantum measurement of the degree of polarization of a light beam," Phys. Rev. Lett. 91, 167902 (2003).
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P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, "New high-intensity source of polarization-entangled photon pairs," Phys. Rev. Lett. 75, 4337-4341 (1995).
[CrossRef] [PubMed]

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C. H. van der Wal, A. C. J. ter Haar, F. K. Wilhelm, R. N. Schouten, C. J. P. M. Harmans, T. P. Orlando, S. Lloyd, and J. E. Mooij, "Quantum superposition of macroscopic persistent-current states," Science 290, 773-777 (2000).
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A. B. Klimov, L. L. Sánchez-Soto, E. C. Yustas, J. Söderholm, and G. Björk, "Distance-based degrees of polarization for a quantum field," Phys. Rev. A 72, 033813 (2005).
[CrossRef]

A. B. Klimov, L. L. Sánchez-Soto, A. Navarro, and E. C. Yustas, "Effective Hamiltonians in quantum optics: a systematic approach," J. Mod. Opt. 49, 2211-2226 (2002).
[CrossRef]

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P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, "New high-intensity source of polarization-entangled photon pairs," Phys. Rev. Lett. 75, 4337-4341 (1995).
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Ann. Phys. (N.Y.) (1)

K. Kraus, "General state changes in quantum theory," Ann. Phys. (N.Y.) 64, 311-335 (1971).
[CrossRef]

Commun. Math. Phys. (1)

G. Lindblad, "On the generators of quantum dynamical semigroups," Commun. Math. Phys. 48, 119-130 (1976).
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Fortschr. Phys. (1)

A. Wünsche, "Quantum-mechanical description of polarized and unpolarized light," Fortschr. Phys. 51, 262-266 (2003).
[CrossRef]

J. Math. Phys. (1)

V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, "Completely positive dynamical semigroups of N-level systems," J. Math. Phys. 17, 821-825 (1976).
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J. Mod. Opt. (2)

A. B. Klimov, L. L. Sánchez-Soto, A. Navarro, and E. C. Yustas, "Effective Hamiltonians in quantum optics: a systematic approach," J. Mod. Opt. 49, 2211-2226 (2002).
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A. P. Alodjants and S. M. Arakelian, "Quantum phase measurements and non-classical polarization states of light," J. Mod. Opt. 46, 475-507 (1999).

J. Opt. B (1)

A. B. Klimov, J. L. Romero, J. Delgado, and L. L. Sánchez-Soto, "Master equations for effective Hamiltonians," J. Opt. B 5, 34-39 (2003).
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Nature (2)

Y. Makhlin, G. Schön, and A. Shnirman, "Josephson-junction qubits with controlled couplings," Nature 398, 305-307 (1999).
[CrossRef]

Y. Nakamura, Yu. A. Pashkin, and J. S. Tsai, "Coherent control of macroscopic quantum states in a single-Cooper-pair box," Nature 398, 786-788 (1999).
[CrossRef]

Opt. Commun. (1)

P. Usachev, J. Söderholm, G. Björk, and A. Trifonov, "Experimental verification of differences between classical and quantum polarization properties," Opt. Commun. 193, 161-173 (2001).
[CrossRef]

Opt. Spectrosc. (1)

J. Söderholm, G. Björk, and A. Trifonov, "Unpolarized light in quantum optics," Opt. Spectrosc. 91, 532-534 (2001).
[CrossRef]

Phys. Lett. A (1)

D. N. Klyshko, "Multiphoton interference and polarization effects," Phys. Lett. A 163, 349-355 (1992).
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Phys. Rev. A (11)

A. B. Klimov, L. L. Sánchez-Soto, E. C. Yustas, J. Söderholm, and G. Björk, "Distance-based degrees of polarization for a quantum field," Phys. Rev. A 72, 033813 (2005).
[CrossRef]

G. A. Durkin, C. Simon, J. Eisert, and D. Bouwmeester, "Resilience of multiphoton entanglement under losses," Phys. Rev. A 70, 062305 (2004).
[CrossRef]

H. Prakash and N. Chandra, "Density operator of unpolarized radiation," Phys. Rev. A 4, 796-799 (1971).
[CrossRef]

J. Lehner, U. Leonhardt, and H. Paul, "Unpolarized light: classical and quantum states," Phys. Rev. A 53, 2727-2735 (1996).
[CrossRef] [PubMed]

M. G. Raymer and A. C. Funk, "Quantum-state tomography of two-mode light using generalized rotations in phase space," Phys. Rev. A 61, 015801 (1999).
[CrossRef]

J. H. Reina, L. Quiroga, and N. F. Johnson, "Decoherence of quantum registers," Phys. Rev. A 65, 032326 (2002).
[CrossRef]

S. G. Schirmer and A. I. Solomon, "Constraints on relaxation rates for N-level quantum systems," Phys. Rev. A 70, 022107 (2004).
[CrossRef]

A. G. Kofman, "Absorption spectrum of an atom strongly coupled to a high-temperature reservoir," Phys. Rev. A 71, 033806 (2005).
[CrossRef]

M. Jakob and S. Stenholm, "Effects of a thermal reservoir on variational functions in open systems," Phys. Rev. A 70, 012104 (2004).
[CrossRef]

A. B. Klimov and L. L. Sánchez-Soto, "Method of small rotations and effective Hamiltonians in nonlinear quantum optics," Phys. Rev. A 61, 063802 (2000).
[CrossRef]

A. B. Klimov, J. L. Romero, and C. Saavedra, "General properties of quantum systems interacting with a field mode in a low-Q cavity," Phys. Rev. A 64, 063802 (2001).
[CrossRef]

Phys. Rev. B (3)

T. Takagahara, "Theory of exciton dephasing in semiconductor quantum dots," Phys. Rev. B 60, 2638-2652 (1999).
[CrossRef]

B. Krummheuer, V. M. Axt, and T. Kuhn, "Theory of pure dephasing and the resulting absorption line shape in semiconductor quantum dots," Phys. Rev. B 65, 195313 (2002).
[CrossRef]

T. Yu and J. H. Eberly, "Phonon decoherence of quantum entanglement: robust and fragile states," Phys. Rev. B 66, 193306 (2002).
[CrossRef]

Phys. Rev. Lett. (7)

A. V. Uskov, A. P. Jauho, B. Tromborg, J. Mørk, and R. Lang, "Dephasing times in quantum dots due to elastic LO phonon-carrier collisions," Phys. Rev. Lett. 85, 1516-1519 (2000).
[CrossRef] [PubMed]

A. O. Caldeira and A. J. Leggett, "Influence of dissipation on quantum tunneling in macroscopic systems," Phys. Rev. Lett. 46, 211-214 (1981).
[CrossRef]

M. Legré, M. Wegmüller, and N. Gisin, "Quantum measurement of the degree of polarization of a light beam," Phys. Rev. Lett. 91, 167902 (2003).
[CrossRef] [PubMed]

A. Aspect, P. Grangier, and G. Roger, "Experimental tests of realistic local theories via Bell's theorem," Phys. Rev. Lett. 47, 460-463 (1981).
[CrossRef]

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, "New high-intensity source of polarization-entangled photon pairs," Phys. Rev. Lett. 75, 4337-4341 (1995).
[CrossRef] [PubMed]

T. Tsegaye, J. Söderholm, M. Atatüre, A. Trifonov, G. Björk, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, "Experimental demonstration of three mutually orthogonal polarization states of entangled photons," Phys. Rev. Lett. 85, 5013-5017 (2000).
[CrossRef] [PubMed]

M. Barbieri, F. De Martini, G. Di Nepi, P. Mataloni, G. M. D'Ariano, and C. Macchiavello, "Detection of entanglement with polarized photons: experimental realization of an entanglement witness," Phys. Rev. Lett. 91, 227901 (2003).
[CrossRef] [PubMed]

Prog. Opt. (1)

A. Luis and L. L. Sánchez-Soto, "Quantum phase difference, phase measurements and quantum Stokes parameters," Prog. Opt. 41, 421-481 (2000).
[CrossRef]

Rev. Mod. Phys. (2)

A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and W. Zwerger, "Dynamics of the dissipative two-state system," Rev. Mod. Phys. 59, 1-85 (1987).
[CrossRef]

A. Galindo and M. A. Martín-Delgado, "Information and computation: classical and quantum aspects," Rev. Mod. Phys. 74, 347-423 (2002).
[CrossRef]

Science (1)

C. H. van der Wal, A. C. J. ter Haar, F. K. Wilhelm, R. N. Schouten, C. J. P. M. Harmans, T. P. Orlando, S. Lloyd, and J. E. Mooij, "Quantum superposition of macroscopic persistent-current states," Science 290, 773-777 (2000).
[CrossRef] [PubMed]

Semicond. Sci. Technol. (1)

E. Pazy, "Calculation of pure dephasing for excitons in quantum dots," Semicond. Sci. Technol. 17, 1172-1179 (2002).
[CrossRef]

Other (7)

To avoid any misunderstanding, we note that what we are calling here single-mode fields are also known as two-mode fields, especially when dealing with polarization properties.

C. W. Gardiner and P. Zoller, Quantum Noise (Springer, 2000).

C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1998).

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

J. M. Jauch and F. Rohrlich, The Theory of Photons and Electrons (Addison-Wesley, 1959).

H. P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford U. Press, 2002).

M. Orszag, Quantum Optics (Springer, 2000).

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Equations (77)

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E ( + ) ( r , t ) = i j = 1 m s = ± ω j 2 ϵ 0 a j s u j s ( r ) exp ( i ω j t ) .
u j s ( r ) = 1 V e j s exp ( i k j r ) ,
[ a j s , a j s ] = δ j j δ s s .
J + = j = 1 m a j + a j , J = j = 1 m a j a j + ,
J z = 1 2 j = 1 m ( a j + a j + a j a j ) .
[ J z , J ± ] = ± J ± , [ J + , J ] = 2 J z .
J 2 = J z 2 + 1 2 ( J + J + J J + ) = N 2 ( N 2 + 1 ) .
N = j = 1 m ( a j + a j + + a j a j )
[ N , J ] = 0 ,
J = 1 2 S
P = S 2 N .
ϱ unpol = N r N 1 N ,
N ( N + 1 ) r N = 1 .
H = H field + H bath + H int .
H field = ω s = ± a s a s ,
ϱ ̇ ( t ) = 1 2 0 d τ Tr B { [ H int ( t ) , [ H int ( t τ ) , ϱ ( t ) ϱ B ] ] } ,
H int = λ s = ± ( κ λ Γ λ a s + κ λ * Γ λ a s ) ,
ϱ ̇ = s γ s 2 L [ a s ] ϱ ,
L [ C s ] ϱ = 2 C s ϱ C s { C s C s , ϱ } ,
H int = λ s = ± ( κ λ Γ λ a s a s + κ λ * Γ λ a s a s ) ,
ϱ ̇ = s = ± γ s 2 L [ a s a s ] ϱ .
+ = 1 + 0 , = 0 + 1 .
ϱ = 1 2 [ 1 + s z s x i s y s x + i s y 1 s z ] = 1 2 ( 1 + s σ ) ,
s x ( t ) = s x ( 0 ) exp [ ( γ + + γ ) t 2 ] ,
s y ( t ) = s y ( 0 ) exp [ ( γ + + γ ) t 2 ] ,
s z ( t ) = s z ( 0 ) .
H sys = H field + H at + V ,
H at = λ 1 2 ω λ σ λ z ,
V = λ s = ± ( g λ s σ λ a s + g λ s * σ λ + a s ) .
H sys ω N + λ 1 2 Ω λ σ λ z + λ , λ λ λ s = ± 1 2 g λ s g λ s * ( 1 Δ λ + 1 Δ λ ) σ λ + σ λ ,
N = s = ± a s a s + λ 1 2 σ λ z
Ω λ = Δ λ + δ λ + J λ Δ λ ,
J λ = 2 g λ 2 [ N + J + exp ( i φ λ ) + J exp ( i φ λ ) ] .
δ λ = g λ 2 Δ λ
ϱ ̇ sys ( t ) = i [ H sys , ϱ sys ] + λ γ λ 2 { ( n ¯ λ + 1 ) L [ σ λ ] ϱ sys + n ¯ λ L [ σ λ + ] ϱ sys } ,
H sys ω N + λ 1 2 Ω λ σ λ z .
ϱ ̇ = i [ ω N , ϱ ] + γ 2 L [ N ] ϱ + γ 2 L [ J + ] ϱ + γ 2 L [ J ] ϱ ,
γ = 4 λ g λ 4 γ λ Δ λ 2 n ¯ λ .
ϱ ( t ) = 1 N + 1 1 .
s x ( t ) = s x ( 0 ) exp ( γ t ) ,
s y ( t ) = s y ( 0 ) exp ( γ t ) ,
s z ( t ) = s z ( 0 ) exp ( 2 γ t ) ,
P ( t ) = [ s x 2 ( 0 ) + s y 2 ( 0 ) + s z 2 ( 0 ) exp ( 2 γ t ) ] 1 2 exp ( γ t ) .
ϱ ̇ = i j = 1 m ω j [ N j , ϱ ] + j = 1 m γ j 2 { L [ N j ] ϱ + L [ J j + ] ϱ + L [ J j ] ϱ } .
ψ = 1 2 ( + 1 2 1 + 2 ) ,
ψ + = 1 2 ( + 1 2 + 1 + 2 ) , + 1 + 2 , 1 2 .
ϱ ( 0 ) = [ ϱ 11 ( 0 ) ϱ 12 ( 0 ) ϱ 13 ( 0 ) ϱ 14 ( 0 ) ϱ 21 ( 0 ) ϱ 22 ( 0 ) ϱ 23 ( 0 ) ϱ 24 ( 0 ) ϱ 31 ( 0 ) ϱ 32 ( 0 ) ϱ 33 ( 0 ) ϱ 34 ( 0 ) ϱ 41 ( 0 ) ϱ 42 ( 0 ) ϱ 43 ( 0 ) ϱ 44 ( 0 ) ] ,
1 = + 1 + 2 , 2 = + 1 2 ,
3 = 1 + 2 , 4 = 1 2 .
ϱ 21 ( t ) = 1 2 { ϱ 21 ( 0 ) [ 1 + exp ( 2 γ 1 t ) ] + ϱ 43 ( 0 ) [ 1 exp ( 2 γ 1 t ) ] } exp ( γ 2 t ) ,
ϱ 31 ( t ) = 1 2 { ϱ 31 ( 0 ) [ 1 + exp ( 2 γ 2 t ) ] + ϱ 42 ( 0 ) [ 1 exp ( 2 γ 2 t ) ] } exp ( γ 1 t ) ,
ϱ 41 ( t ) = ϱ 41 ( 0 ) exp [ ( γ 1 + γ 2 ) t ] ,
ϱ 32 ( t ) = ϱ 32 ( 0 ) exp [ ( γ 1 + γ 2 ) t ] ,
ϱ 42 ( t ) = 1 2 { ϱ 42 ( 0 ) [ 1 + exp ( 2 γ 2 t ) ] + ϱ 31 ( 0 ) [ 1 exp ( 2 γ 2 t ) ] } exp ( γ 1 t ) ,
ϱ 43 ( t ) = 1 2 { ϱ 43 ( 0 ) [ 1 + exp ( 2 γ 1 t ) ] + ϱ 21 ( 0 ) [ 1 exp ( 2 γ 1 t ) ] } exp ( γ 2 t )
ϱ 11 ( t ) = 1 4 ( 1 + { 2 [ ϱ 11 ( 0 ) + ϱ 22 ( 0 ) ] 1 } exp ( 2 γ 1 t ) + { 2 [ ϱ 11 ( 0 ) + ϱ 33 ( 0 ) ] 1 } exp ( 2 γ 2 t ) + { 2 [ ϱ 11 ( 0 ) + ϱ 44 ( 0 ) ] 1 } exp [ 2 ( γ 1 + γ 2 ) t ] ) ,
ϱ 22 ( t ) = 1 4 ( 1 + { 2 [ ϱ 22 ( 0 ) + ϱ 11 ( 0 ) ] 1 } exp ( 2 γ 1 t ) + { 2 [ ϱ 22 ( 0 ) + ϱ 44 ( 0 ) ] 1 } exp ( 2 γ 2 t ) + { 2 [ ϱ 22 ( 0 ) + ϱ 33 ( 0 ) ] 1 } exp [ 2 ( γ 1 + γ 2 ) t ] ) ,
ϱ 33 ( t ) = 1 4 ( 1 + { 2 [ ϱ 33 ( 0 ) + ϱ 44 ( 0 ) ] 1 } exp ( 2 γ 1 t ) + { 2 [ ϱ 33 ( 0 ) + ϱ 11 ( 0 ) ] 1 } exp ( 2 γ 2 t ) + { 2 [ ϱ 33 ( 0 ) + ϱ 22 ( 0 ) ] 1 } exp [ 2 ( γ 1 + γ 2 ) t ] )
J x = 1 2 [ 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 ] , J y = 1 2 [ 0 i i 0 i 0 0 i i 0 0 i 0 i i 0 ] ,
J z = [ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 ] .
P ( t ) = [ ϱ 12 ( t ) + ϱ 13 ( t ) + ϱ 24 ( t ) + ϱ 34 ( t ) 2 + ϱ 11 ( t ) ϱ 44 ( t ) 2 ] .
P ( t ) = 1 2 [ exp ( 2 γ 1 t ) + exp ( 2 γ 2 t ) ] .
ψ ± ψ ± = 1 2 [ 0 0 0 0 0 1 ± 1 0 0 ± 1 1 0 0 0 0 0 ] ,
P ( t ) = 1 2 exp [ ( γ 1 + γ 2 ) t ] .
N λ + ϱ = σ λ + ϱ σ λ , N λ ϱ = σ λ ϱ σ λ + ,
N λ l ϱ = 1 2 σ λ z ϱ , N λ r ϱ = 1 2 ϱ σ λ z ,
A λ l ϱ = Ω λ ϱ , A λ r ϱ = ϱ Ω λ ,
[ N λ ± , N λ l ] = N λ ± δ λ λ ,
[ N λ ± , N λ r ] = N λ ± δ λ λ ,
[ N λ α , A λ β ] = 0 ,
̇ sys = λ γ λ n ¯ λ ϱ sys + λ [ γ λ n ¯ λ ( N λ + + N λ ) 2 i A λ l N λ r + 2 i A λ r N λ r ] ϱ sys .
V = exp [ π 2 λ 1 2 ( N λ + N λ ) ] ,
ϱ ̇ sys = λ [ γ λ n ¯ λ + γ λ n ¯ λ ( N λ r + N λ l ) 2 i ( A λ r A λ l ) ( N λ + + N λ ) + i ( A λ r + A λ l ) ( N λ l N λ r ) ] ϱ sys .
U = exp [ i λ A λ r A λ l 2 γ λ n ¯ λ ( N λ + N λ ) ] .
ϱ ̇ sys = λ { γ λ n ¯ λ [ 1 N λ + N λ ] + i ( A λ r + A λ l ) ( N λ l N λ r ) } ϱ sys λ 2 γ λ n ¯ λ ( A λ r A λ l 2 γ λ n ¯ λ ) 2 ( N λ + + N λ ) ϱ sys ,
ϱ ̇ = λ 1 2 γ λ Δ λ n ¯ λ L [ J λ ] ϱ ,
ϱ ̇ = γ 2 L [ N ] ϱ + γ 2 L [ J + ] ϱ + γ 2 L [ J ] ϱ ,

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