Abstract

We investigate how the squeezed thermal state (STS) and squeezed number state (SNS) evolve undergoing decoherence in the laser process. Remarkably, the initial SNS, an example of a pure state, evolves into a mixed state, which turns out to be a Laguerre polynomial of combination of creation and annihilation operators within normal ordering; however, the STS, a mixed state, still keeps squeezed and thermal. At long times, these fields lose their nonclassical nature and decay to a highly classical thermal field. The normally ordered density operators of such states in the laser channel lead to deriving the analytical time-evolution expressions of the Wigner functions (WFs). Their nonclassicality is investigated in reference to the time-evolution WFs, which indicates that both of the WFs decay to the same thermal WF as a result of decoherence when the decay time t or the cavity loss κ of the laser is large enough.

© 2012 Optical Society of America

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  1. A. Vourdas and R. M. Weiner, “Photon-counting distribution in squeezed states,” Phys. Rev. A 36, 5866–5869 (1987).
    [CrossRef]
  2. V. V. Dodonov, O. V. Man’ko, V. I. Man’ko, and L. Rosa, “Thermal noise and oscillations of the photon distribution for squeezed and correlated light,” Phys. Lett. A 185, 231–237 (1994).
    [CrossRef]
  3. Z. H. Musslimani, S. L. Braunstein, A. Mann, and M. Revzen, “Destruction of photocount oscillations by thermal noise,” Phys. Rev. A 51, 4967–4973 (1995).
    [CrossRef]
  4. P. Marian and T. A. Marian, “Squeezed states with thermal noise. I. Photon-number statistics,” Phys. Rev. A 47, 4474–4486 (1993).
    [CrossRef]
  5. P. Marian and T. A. Marian, “Squeezed states with thermal noise. II. Damping and photon counting,” Phys. Rev. A 47, 4487–4495 (1993).
    [CrossRef]
  6. P. Marian, T. A. Marian, and H. Scutaru, “Quantifying nonclassicality of one-mode Gaussian states of the radiation field,” Phys. Rev. Lett. 88, 153601 (2002).
    [CrossRef]
  7. M. S. Kim and V. Bužek, “Schrödinger-cat states at finite temperature: influence of a finite-temperature heat bath on quantum interferences,” Phys. Rev. A 46, 4239–4251 (1992).
    [CrossRef]
  8. A. Biswas and G. S. Agarwal, “Nonclassicality and decoherence of photon-subtracted squeezed states,” Phys. Rev. A 75, 032104 (2007).
    [CrossRef]
  9. L. Y. Hu and H. Y. Fan, “Statistical properties of photon-subtracted squeezed vacuum in thermal environment,” J. Opt. Soc. Am. B 25, 1955–1964 (2008).
    [CrossRef]
  10. T. Hiroshima, “Decoherence and entanglement in two-mode squeezed vacuum states,” Phys. Rev. A 63, 022305 (2001).
    [CrossRef]
  11. H. Jeong, J. Lee, and M. S. Kim, “Dynamics of nonlocality for a two-mode squeezed state in a thermal environment,” Phys. Rev. A 61, 052101 (2000).
    [CrossRef]
  12. J. S. Prauzner-Bechcicki, “Two-mode squeezed vacuum state coupled to the common thermal reservoir,” J. Phys. A 37, L173–L181 (2004).
    [CrossRef]
  13. C. Gardiner and P. Zoller, Quantum Noise (Springer, 2000).
  14. P. Marian, “Higher-order squeezing properties and correlation functions for squeezed number states,” Phys. Rev. A 44, 3325–3330 (1991).
    [CrossRef]
  15. P. Marian, “Higher-order squeezing and photon statistics for squeezed thermal states,” Phys. Rev. A 45, 2044–2051 (1992).
    [CrossRef]
  16. M. S. Kim, F. A. M. de Oliveira, and P. L. Knight, “Properties of squeezed number states and squeezed thermal states,” Phys. Rev. A 40, 2494–2503 (1989).
    [CrossRef]
  17. M. S. Kim and V. Bužek, “Photon statistics of superposition states in phase-sensitive reservoirs,”Phys. Rev. A 47, 610–619 (1993).
    [CrossRef]
  18. P. Marian and T. A. Marian, “Destruction of higher-order squeezing by thermal noise,” J. Phys. A 29, 6233–6245 (1996).
    [CrossRef]
  19. Z. Cheng, “Quantum effects of thermal radiation in a Kerr nonlinear blackbody,” J. Opt. Soc. Am. B 19, 1692–1705 (2002).
    [CrossRef]
  20. Gh.-S. Paraoanu and H. Scutaru, “Fidelity for multimode thermal squeezed states,” Phys. Rev. A 61, 022306 (2000).
    [CrossRef]
  21. P. Marian, T. A. Marian, and H. Scutaru, “Bures distance as a measure of entanglement for two-mode squeezed thermal states,” Phys. Rev. A 68, 062309 (2003).
    [CrossRef]
  22. J. Anders, “Thermal state entanglement in harmonic lattices,” Phys. Rev. A 77, 062102 (2008).
    [CrossRef]
  23. X. B. Wang, C. H. Oh, and L. C. Kwek, “Bures fidelity of displaced squeezed thermal states,” Phys. Rev. A 58, 4186–4190(1998).
    [CrossRef]
  24. M. Aspachs, J. Calsamiglia, R. Munoz-Tapia, and E. Bagan, “Phase estimation for thermal Gaussian states,” Phys. Rev. A 79, 033834 (2009).
    [CrossRef]
  25. A. V. Chizhov, T. Gantsogt, and B. K. Murzakhmetovt, “Phase distributions of squeezed number states and squeezed thermal states,” Quantum Opt. 5, 85–93 (1993).
    [CrossRef]
  26. H. Y. Fan and L. Y. Hu, “Operator-sum representation of density operators as solution to master equations obtained via the entangled states approach,” Mod. Phys. Lett. B 22, 2435–2468(2008).
    [CrossRef]
  27. H. Y. Fan, Representation and Transformation Theory in Quantum Mechanics (Shanghai Scientific and Technical, 1997), in Chinese.
  28. M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University, 2004).
  29. R. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131, 2766–2788 (1963).
    [CrossRef]
  30. J. R. Klauder and B. S. Skargerstam, Coherent States (World Scientific, 1985).
  31. T. Kiesel, W. Vogel, M. Bellini, and A. Zavatta, “Nonclassicality quasiprobability of single-photon-added thermal states,” Phys. Rev. A 83, 032116 (2011).
    [CrossRef]
  32. H. Y. Fan and G. Ren, “Evolution of number state to density operator of binomial distribution in the amplitude dissipative channel,” Chin. Phys. Lett. 27, 050302 (2010).
    [CrossRef]
  33. H. Y. Fan, “Application of IWOP technique to the generalized Weyl correspondence,” Phys. Lett. A 124, 303–307 (1987).
    [CrossRef]
  34. H. Y. Fan, “Newton–Leibniz integration for ket–bra operators in quantum mechanics (IV)—integrations within Weyl ordered product of operators and their applications,” Ann. Phys. 323, 500–526 (2008).
    [CrossRef]
  35. H. Weyl, The Classical Groups (Princeton University, 1953).
  36. W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, 1973).
  37. C. L. Methta, “Diagonal coherent-state representation of quantum operators,” Phys. Rev. Lett. 18, 752–754 (1967).
    [CrossRef]

2011 (1)

T. Kiesel, W. Vogel, M. Bellini, and A. Zavatta, “Nonclassicality quasiprobability of single-photon-added thermal states,” Phys. Rev. A 83, 032116 (2011).
[CrossRef]

2010 (1)

H. Y. Fan and G. Ren, “Evolution of number state to density operator of binomial distribution in the amplitude dissipative channel,” Chin. Phys. Lett. 27, 050302 (2010).
[CrossRef]

2009 (1)

M. Aspachs, J. Calsamiglia, R. Munoz-Tapia, and E. Bagan, “Phase estimation for thermal Gaussian states,” Phys. Rev. A 79, 033834 (2009).
[CrossRef]

2008 (4)

H. Y. Fan and L. Y. Hu, “Operator-sum representation of density operators as solution to master equations obtained via the entangled states approach,” Mod. Phys. Lett. B 22, 2435–2468(2008).
[CrossRef]

J. Anders, “Thermal state entanglement in harmonic lattices,” Phys. Rev. A 77, 062102 (2008).
[CrossRef]

H. Y. Fan, “Newton–Leibniz integration for ket–bra operators in quantum mechanics (IV)—integrations within Weyl ordered product of operators and their applications,” Ann. Phys. 323, 500–526 (2008).
[CrossRef]

L. Y. Hu and H. Y. Fan, “Statistical properties of photon-subtracted squeezed vacuum in thermal environment,” J. Opt. Soc. Am. B 25, 1955–1964 (2008).
[CrossRef]

2007 (1)

A. Biswas and G. S. Agarwal, “Nonclassicality and decoherence of photon-subtracted squeezed states,” Phys. Rev. A 75, 032104 (2007).
[CrossRef]

2004 (1)

J. S. Prauzner-Bechcicki, “Two-mode squeezed vacuum state coupled to the common thermal reservoir,” J. Phys. A 37, L173–L181 (2004).
[CrossRef]

2003 (1)

P. Marian, T. A. Marian, and H. Scutaru, “Bures distance as a measure of entanglement for two-mode squeezed thermal states,” Phys. Rev. A 68, 062309 (2003).
[CrossRef]

2002 (2)

Z. Cheng, “Quantum effects of thermal radiation in a Kerr nonlinear blackbody,” J. Opt. Soc. Am. B 19, 1692–1705 (2002).
[CrossRef]

P. Marian, T. A. Marian, and H. Scutaru, “Quantifying nonclassicality of one-mode Gaussian states of the radiation field,” Phys. Rev. Lett. 88, 153601 (2002).
[CrossRef]

2001 (1)

T. Hiroshima, “Decoherence and entanglement in two-mode squeezed vacuum states,” Phys. Rev. A 63, 022305 (2001).
[CrossRef]

2000 (2)

H. Jeong, J. Lee, and M. S. Kim, “Dynamics of nonlocality for a two-mode squeezed state in a thermal environment,” Phys. Rev. A 61, 052101 (2000).
[CrossRef]

Gh.-S. Paraoanu and H. Scutaru, “Fidelity for multimode thermal squeezed states,” Phys. Rev. A 61, 022306 (2000).
[CrossRef]

1998 (1)

X. B. Wang, C. H. Oh, and L. C. Kwek, “Bures fidelity of displaced squeezed thermal states,” Phys. Rev. A 58, 4186–4190(1998).
[CrossRef]

1996 (1)

P. Marian and T. A. Marian, “Destruction of higher-order squeezing by thermal noise,” J. Phys. A 29, 6233–6245 (1996).
[CrossRef]

1995 (1)

Z. H. Musslimani, S. L. Braunstein, A. Mann, and M. Revzen, “Destruction of photocount oscillations by thermal noise,” Phys. Rev. A 51, 4967–4973 (1995).
[CrossRef]

1994 (1)

V. V. Dodonov, O. V. Man’ko, V. I. Man’ko, and L. Rosa, “Thermal noise and oscillations of the photon distribution for squeezed and correlated light,” Phys. Lett. A 185, 231–237 (1994).
[CrossRef]

1993 (4)

P. Marian and T. A. Marian, “Squeezed states with thermal noise. I. Photon-number statistics,” Phys. Rev. A 47, 4474–4486 (1993).
[CrossRef]

P. Marian and T. A. Marian, “Squeezed states with thermal noise. II. Damping and photon counting,” Phys. Rev. A 47, 4487–4495 (1993).
[CrossRef]

M. S. Kim and V. Bužek, “Photon statistics of superposition states in phase-sensitive reservoirs,”Phys. Rev. A 47, 610–619 (1993).
[CrossRef]

A. V. Chizhov, T. Gantsogt, and B. K. Murzakhmetovt, “Phase distributions of squeezed number states and squeezed thermal states,” Quantum Opt. 5, 85–93 (1993).
[CrossRef]

1992 (2)

P. Marian, “Higher-order squeezing and photon statistics for squeezed thermal states,” Phys. Rev. A 45, 2044–2051 (1992).
[CrossRef]

M. S. Kim and V. Bužek, “Schrödinger-cat states at finite temperature: influence of a finite-temperature heat bath on quantum interferences,” Phys. Rev. A 46, 4239–4251 (1992).
[CrossRef]

1991 (1)

P. Marian, “Higher-order squeezing properties and correlation functions for squeezed number states,” Phys. Rev. A 44, 3325–3330 (1991).
[CrossRef]

1989 (1)

M. S. Kim, F. A. M. de Oliveira, and P. L. Knight, “Properties of squeezed number states and squeezed thermal states,” Phys. Rev. A 40, 2494–2503 (1989).
[CrossRef]

1987 (2)

A. Vourdas and R. M. Weiner, “Photon-counting distribution in squeezed states,” Phys. Rev. A 36, 5866–5869 (1987).
[CrossRef]

H. Y. Fan, “Application of IWOP technique to the generalized Weyl correspondence,” Phys. Lett. A 124, 303–307 (1987).
[CrossRef]

1967 (1)

C. L. Methta, “Diagonal coherent-state representation of quantum operators,” Phys. Rev. Lett. 18, 752–754 (1967).
[CrossRef]

1963 (1)

R. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131, 2766–2788 (1963).
[CrossRef]

Agarwal, G. S.

A. Biswas and G. S. Agarwal, “Nonclassicality and decoherence of photon-subtracted squeezed states,” Phys. Rev. A 75, 032104 (2007).
[CrossRef]

Anders, J.

J. Anders, “Thermal state entanglement in harmonic lattices,” Phys. Rev. A 77, 062102 (2008).
[CrossRef]

Aspachs, M.

M. Aspachs, J. Calsamiglia, R. Munoz-Tapia, and E. Bagan, “Phase estimation for thermal Gaussian states,” Phys. Rev. A 79, 033834 (2009).
[CrossRef]

Bagan, E.

M. Aspachs, J. Calsamiglia, R. Munoz-Tapia, and E. Bagan, “Phase estimation for thermal Gaussian states,” Phys. Rev. A 79, 033834 (2009).
[CrossRef]

Bellini, M.

T. Kiesel, W. Vogel, M. Bellini, and A. Zavatta, “Nonclassicality quasiprobability of single-photon-added thermal states,” Phys. Rev. A 83, 032116 (2011).
[CrossRef]

Biswas, A.

A. Biswas and G. S. Agarwal, “Nonclassicality and decoherence of photon-subtracted squeezed states,” Phys. Rev. A 75, 032104 (2007).
[CrossRef]

Braunstein, S. L.

Z. H. Musslimani, S. L. Braunstein, A. Mann, and M. Revzen, “Destruction of photocount oscillations by thermal noise,” Phys. Rev. A 51, 4967–4973 (1995).
[CrossRef]

Bužek, V.

M. S. Kim and V. Bužek, “Photon statistics of superposition states in phase-sensitive reservoirs,”Phys. Rev. A 47, 610–619 (1993).
[CrossRef]

M. S. Kim and V. Bužek, “Schrödinger-cat states at finite temperature: influence of a finite-temperature heat bath on quantum interferences,” Phys. Rev. A 46, 4239–4251 (1992).
[CrossRef]

Calsamiglia, J.

M. Aspachs, J. Calsamiglia, R. Munoz-Tapia, and E. Bagan, “Phase estimation for thermal Gaussian states,” Phys. Rev. A 79, 033834 (2009).
[CrossRef]

Cheng, Z.

Chizhov, A. V.

A. V. Chizhov, T. Gantsogt, and B. K. Murzakhmetovt, “Phase distributions of squeezed number states and squeezed thermal states,” Quantum Opt. 5, 85–93 (1993).
[CrossRef]

de Oliveira, F. A. M.

M. S. Kim, F. A. M. de Oliveira, and P. L. Knight, “Properties of squeezed number states and squeezed thermal states,” Phys. Rev. A 40, 2494–2503 (1989).
[CrossRef]

Dodonov, V. V.

V. V. Dodonov, O. V. Man’ko, V. I. Man’ko, and L. Rosa, “Thermal noise and oscillations of the photon distribution for squeezed and correlated light,” Phys. Lett. A 185, 231–237 (1994).
[CrossRef]

Fan, H. Y.

H. Y. Fan and G. Ren, “Evolution of number state to density operator of binomial distribution in the amplitude dissipative channel,” Chin. Phys. Lett. 27, 050302 (2010).
[CrossRef]

H. Y. Fan, “Newton–Leibniz integration for ket–bra operators in quantum mechanics (IV)—integrations within Weyl ordered product of operators and their applications,” Ann. Phys. 323, 500–526 (2008).
[CrossRef]

H. Y. Fan and L. Y. Hu, “Operator-sum representation of density operators as solution to master equations obtained via the entangled states approach,” Mod. Phys. Lett. B 22, 2435–2468(2008).
[CrossRef]

L. Y. Hu and H. Y. Fan, “Statistical properties of photon-subtracted squeezed vacuum in thermal environment,” J. Opt. Soc. Am. B 25, 1955–1964 (2008).
[CrossRef]

H. Y. Fan, “Application of IWOP technique to the generalized Weyl correspondence,” Phys. Lett. A 124, 303–307 (1987).
[CrossRef]

H. Y. Fan, Representation and Transformation Theory in Quantum Mechanics (Shanghai Scientific and Technical, 1997), in Chinese.

Gantsogt, T.

A. V. Chizhov, T. Gantsogt, and B. K. Murzakhmetovt, “Phase distributions of squeezed number states and squeezed thermal states,” Quantum Opt. 5, 85–93 (1993).
[CrossRef]

Gardiner, C.

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

Glauber, R.

R. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131, 2766–2788 (1963).
[CrossRef]

Hiroshima, T.

T. Hiroshima, “Decoherence and entanglement in two-mode squeezed vacuum states,” Phys. Rev. A 63, 022305 (2001).
[CrossRef]

Hu, L. Y.

L. Y. Hu and H. Y. Fan, “Statistical properties of photon-subtracted squeezed vacuum in thermal environment,” J. Opt. Soc. Am. B 25, 1955–1964 (2008).
[CrossRef]

H. Y. Fan and L. Y. Hu, “Operator-sum representation of density operators as solution to master equations obtained via the entangled states approach,” Mod. Phys. Lett. B 22, 2435–2468(2008).
[CrossRef]

Jeong, H.

H. Jeong, J. Lee, and M. S. Kim, “Dynamics of nonlocality for a two-mode squeezed state in a thermal environment,” Phys. Rev. A 61, 052101 (2000).
[CrossRef]

Kiesel, T.

T. Kiesel, W. Vogel, M. Bellini, and A. Zavatta, “Nonclassicality quasiprobability of single-photon-added thermal states,” Phys. Rev. A 83, 032116 (2011).
[CrossRef]

Kim, M. S.

H. Jeong, J. Lee, and M. S. Kim, “Dynamics of nonlocality for a two-mode squeezed state in a thermal environment,” Phys. Rev. A 61, 052101 (2000).
[CrossRef]

M. S. Kim and V. Bužek, “Photon statistics of superposition states in phase-sensitive reservoirs,”Phys. Rev. A 47, 610–619 (1993).
[CrossRef]

M. S. Kim and V. Bužek, “Schrödinger-cat states at finite temperature: influence of a finite-temperature heat bath on quantum interferences,” Phys. Rev. A 46, 4239–4251 (1992).
[CrossRef]

M. S. Kim, F. A. M. de Oliveira, and P. L. Knight, “Properties of squeezed number states and squeezed thermal states,” Phys. Rev. A 40, 2494–2503 (1989).
[CrossRef]

Klauder, J. R.

J. R. Klauder and B. S. Skargerstam, Coherent States (World Scientific, 1985).

Knight, P. L.

M. S. Kim, F. A. M. de Oliveira, and P. L. Knight, “Properties of squeezed number states and squeezed thermal states,” Phys. Rev. A 40, 2494–2503 (1989).
[CrossRef]

Kwek, L. C.

X. B. Wang, C. H. Oh, and L. C. Kwek, “Bures fidelity of displaced squeezed thermal states,” Phys. Rev. A 58, 4186–4190(1998).
[CrossRef]

Lee, J.

H. Jeong, J. Lee, and M. S. Kim, “Dynamics of nonlocality for a two-mode squeezed state in a thermal environment,” Phys. Rev. A 61, 052101 (2000).
[CrossRef]

Louisell, W. H.

W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, 1973).

Man’ko, O. V.

V. V. Dodonov, O. V. Man’ko, V. I. Man’ko, and L. Rosa, “Thermal noise and oscillations of the photon distribution for squeezed and correlated light,” Phys. Lett. A 185, 231–237 (1994).
[CrossRef]

Man’ko, V. I.

V. V. Dodonov, O. V. Man’ko, V. I. Man’ko, and L. Rosa, “Thermal noise and oscillations of the photon distribution for squeezed and correlated light,” Phys. Lett. A 185, 231–237 (1994).
[CrossRef]

Mann, A.

Z. H. Musslimani, S. L. Braunstein, A. Mann, and M. Revzen, “Destruction of photocount oscillations by thermal noise,” Phys. Rev. A 51, 4967–4973 (1995).
[CrossRef]

Marian, P.

P. Marian, T. A. Marian, and H. Scutaru, “Bures distance as a measure of entanglement for two-mode squeezed thermal states,” Phys. Rev. A 68, 062309 (2003).
[CrossRef]

P. Marian, T. A. Marian, and H. Scutaru, “Quantifying nonclassicality of one-mode Gaussian states of the radiation field,” Phys. Rev. Lett. 88, 153601 (2002).
[CrossRef]

P. Marian and T. A. Marian, “Destruction of higher-order squeezing by thermal noise,” J. Phys. A 29, 6233–6245 (1996).
[CrossRef]

P. Marian and T. A. Marian, “Squeezed states with thermal noise. I. Photon-number statistics,” Phys. Rev. A 47, 4474–4486 (1993).
[CrossRef]

P. Marian and T. A. Marian, “Squeezed states with thermal noise. II. Damping and photon counting,” Phys. Rev. A 47, 4487–4495 (1993).
[CrossRef]

P. Marian, “Higher-order squeezing and photon statistics for squeezed thermal states,” Phys. Rev. A 45, 2044–2051 (1992).
[CrossRef]

P. Marian, “Higher-order squeezing properties and correlation functions for squeezed number states,” Phys. Rev. A 44, 3325–3330 (1991).
[CrossRef]

Marian, T. A.

P. Marian, T. A. Marian, and H. Scutaru, “Bures distance as a measure of entanglement for two-mode squeezed thermal states,” Phys. Rev. A 68, 062309 (2003).
[CrossRef]

P. Marian, T. A. Marian, and H. Scutaru, “Quantifying nonclassicality of one-mode Gaussian states of the radiation field,” Phys. Rev. Lett. 88, 153601 (2002).
[CrossRef]

P. Marian and T. A. Marian, “Destruction of higher-order squeezing by thermal noise,” J. Phys. A 29, 6233–6245 (1996).
[CrossRef]

P. Marian and T. A. Marian, “Squeezed states with thermal noise. II. Damping and photon counting,” Phys. Rev. A 47, 4487–4495 (1993).
[CrossRef]

P. Marian and T. A. Marian, “Squeezed states with thermal noise. I. Photon-number statistics,” Phys. Rev. A 47, 4474–4486 (1993).
[CrossRef]

Methta, C. L.

C. L. Methta, “Diagonal coherent-state representation of quantum operators,” Phys. Rev. Lett. 18, 752–754 (1967).
[CrossRef]

Munoz-Tapia, R.

M. Aspachs, J. Calsamiglia, R. Munoz-Tapia, and E. Bagan, “Phase estimation for thermal Gaussian states,” Phys. Rev. A 79, 033834 (2009).
[CrossRef]

Murzakhmetovt, B. K.

A. V. Chizhov, T. Gantsogt, and B. K. Murzakhmetovt, “Phase distributions of squeezed number states and squeezed thermal states,” Quantum Opt. 5, 85–93 (1993).
[CrossRef]

Musslimani, Z. H.

Z. H. Musslimani, S. L. Braunstein, A. Mann, and M. Revzen, “Destruction of photocount oscillations by thermal noise,” Phys. Rev. A 51, 4967–4973 (1995).
[CrossRef]

Oh, C. H.

X. B. Wang, C. H. Oh, and L. C. Kwek, “Bures fidelity of displaced squeezed thermal states,” Phys. Rev. A 58, 4186–4190(1998).
[CrossRef]

Paraoanu, Gh.-S.

Gh.-S. Paraoanu and H. Scutaru, “Fidelity for multimode thermal squeezed states,” Phys. Rev. A 61, 022306 (2000).
[CrossRef]

Prauzner-Bechcicki, J. S.

J. S. Prauzner-Bechcicki, “Two-mode squeezed vacuum state coupled to the common thermal reservoir,” J. Phys. A 37, L173–L181 (2004).
[CrossRef]

Ren, G.

H. Y. Fan and G. Ren, “Evolution of number state to density operator of binomial distribution in the amplitude dissipative channel,” Chin. Phys. Lett. 27, 050302 (2010).
[CrossRef]

Revzen, M.

Z. H. Musslimani, S. L. Braunstein, A. Mann, and M. Revzen, “Destruction of photocount oscillations by thermal noise,” Phys. Rev. A 51, 4967–4973 (1995).
[CrossRef]

Rosa, L.

V. V. Dodonov, O. V. Man’ko, V. I. Man’ko, and L. Rosa, “Thermal noise and oscillations of the photon distribution for squeezed and correlated light,” Phys. Lett. A 185, 231–237 (1994).
[CrossRef]

Scully, M. O.

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University, 2004).

Scutaru, H.

P. Marian, T. A. Marian, and H. Scutaru, “Bures distance as a measure of entanglement for two-mode squeezed thermal states,” Phys. Rev. A 68, 062309 (2003).
[CrossRef]

P. Marian, T. A. Marian, and H. Scutaru, “Quantifying nonclassicality of one-mode Gaussian states of the radiation field,” Phys. Rev. Lett. 88, 153601 (2002).
[CrossRef]

Gh.-S. Paraoanu and H. Scutaru, “Fidelity for multimode thermal squeezed states,” Phys. Rev. A 61, 022306 (2000).
[CrossRef]

Skargerstam, B. S.

J. R. Klauder and B. S. Skargerstam, Coherent States (World Scientific, 1985).

Vogel, W.

T. Kiesel, W. Vogel, M. Bellini, and A. Zavatta, “Nonclassicality quasiprobability of single-photon-added thermal states,” Phys. Rev. A 83, 032116 (2011).
[CrossRef]

Vourdas, A.

A. Vourdas and R. M. Weiner, “Photon-counting distribution in squeezed states,” Phys. Rev. A 36, 5866–5869 (1987).
[CrossRef]

Wang, X. B.

X. B. Wang, C. H. Oh, and L. C. Kwek, “Bures fidelity of displaced squeezed thermal states,” Phys. Rev. A 58, 4186–4190(1998).
[CrossRef]

Weiner, R. M.

A. Vourdas and R. M. Weiner, “Photon-counting distribution in squeezed states,” Phys. Rev. A 36, 5866–5869 (1987).
[CrossRef]

Weyl, H.

H. Weyl, The Classical Groups (Princeton University, 1953).

Zavatta, A.

T. Kiesel, W. Vogel, M. Bellini, and A. Zavatta, “Nonclassicality quasiprobability of single-photon-added thermal states,” Phys. Rev. A 83, 032116 (2011).
[CrossRef]

Zoller, P.

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

Zubairy, M. S.

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University, 2004).

Ann. Phys. (1)

H. Y. Fan, “Newton–Leibniz integration for ket–bra operators in quantum mechanics (IV)—integrations within Weyl ordered product of operators and their applications,” Ann. Phys. 323, 500–526 (2008).
[CrossRef]

Chin. Phys. Lett. (1)

H. Y. Fan and G. Ren, “Evolution of number state to density operator of binomial distribution in the amplitude dissipative channel,” Chin. Phys. Lett. 27, 050302 (2010).
[CrossRef]

J. Opt. Soc. Am. B (2)

J. Phys. A (2)

J. S. Prauzner-Bechcicki, “Two-mode squeezed vacuum state coupled to the common thermal reservoir,” J. Phys. A 37, L173–L181 (2004).
[CrossRef]

P. Marian and T. A. Marian, “Destruction of higher-order squeezing by thermal noise,” J. Phys. A 29, 6233–6245 (1996).
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Figures (2)

Fig. 1.
Fig. 1.

Evolution of WF distributions of the SNS with given g=1 for (a) n=0, γ=0.3, κ=2, t=0.01; (b) n=3, γ=0, κ=2, t=0.01; (c) n=3, γ=0.3, κ=2, t=0.01; (d) n=6, γ=0.3, κ=2, t=0.01; (e) n=3, γ=0.3, κ=2, t=0.08; and (f) n=3, γ=0.3, κ=12, t=0.01.

Fig. 2.
Fig. 2.

Evolution of WF distributions of the STS with given γ=0.5, n¯=1, and g=1 for (a) κ=2, t=0.01; (b) κ=2, t=5; and (c) κ=200, t=0.01.

Equations (61)

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dρ(t)dt=g[2aρ(t)aaaρ(t)ρ(t)aa]+κ[2aρ(t)aaaρ(t)ρ(t)aa],
|η=exp(12|η|2+ηaη*a˜+aa˜)|00˜=D(η)|η=0,
a|η=0=a˜|η=0,a|η=0=a˜|η=0,aa|η=0=a˜a˜|η=0.
ddt|ρ(t)={g[2aρ(t)aaaρ(t)ρ(t)aa]+κ[2aρ(t)aaaρ(t)ρ(t)aa]}|I={g[2aa˜aaa˜a˜]+κ[2aa˜aaa˜a˜]}|ρ(t),
|ρ(t)=exp{gt[2aa˜aaa˜a˜]+κt(2aa˜aaa˜a˜)}|ρ0,
[aa˜a˜a,(aa˜)(aa˜)]=2[(aa˜)(aa˜)]
|ρ(t)=exp[(aa˜a˜a˜+1)(κg)t]exp{(κ+g)[1e2(κg)t]2(κg)(aa˜)(aa˜)}|ρ0.
|ρ(t)=(1gT1)exp(gT1aa˜)exp{(a˜a˜+aa)lnT2}exp(κT1aa˜)ρ0|I=(1gT1)i,j=0κigji!j!T1i+jajeaalnT2aiρ0aieaalnT2aj|I,
ρ(t)=(1gT1)i,j=0κigji!j!T1i+jajeaalnT2aiρ0aieaalnT2aj,
ρ(t)=n=0Mnρ0Mn,
Mi,j(1gT1)κigji!j!T1i+jajeaalnT2ai.
ρsn=S(γ)ρnS1(γ),
eγ(a2a2)/2an=eγ(a2a2)/2d2zπ|zz|an=sech1/2γd2zπ:exp{|z|2+zasechγ+z*a+z22tanhγa22tanhγaa}z*n:.
eγ(a2a2)/2an=sech1/2γ:exp[tanhγ2(a2a2)+(sechγ1)aa]×k=0[n/2]n!k!(n2k)!(tanhγ2)k(asechγ+atanhγ)n2k:.
S(γ)|n=sechγ(tanhγ)n2nn!Hn(asinh2γ)exp(tanhγ2a2)|0,
ρsn=sechγ(tanhγ)n2nn!:Hn(asinh2γ)Hn(asinh2γ)×exp[tanhγ2(a2+a2)aa]:.
ρsn(t)=(1gT1)sechγ(tanhγ)n2nn!×i,j=0κigji!j!T1i+jajeaalnT2ai:Hn(asinh2γ)×Hn(asinh2γ)×exp[tanhγ2(a2+a2)aa]:aieaalnT2aj.
eaalnT2|z=e|z|2/2+zT2a|0
ρsn(t)=ABC2nl=0n(n!)2l![(nl)!]2(4BκT1C2sinh2γ)l×:Hnl[BT2(aκT1atanhγ)Csinh2γ]Hnl[BT2(aκT1atanhγ)Csinh2γ]eΩ:,
A=(1gT1)(tanhγ)nsechγ2nn!,B=11κ2T12tanh2γ,C=B(1κ2T12),
eΩexp[BT22tanhγ2(a2+a2)+(BκT1T22tanh2γ+gT11)aa]:.
:Hnl[BT2(aκT1atanhγ)Csinh2γ]Hnl[BT2(aκT1atanhγ)Csinh2γ]eΩ:=::Hnl[BT2(aκT1atanhγ)Csinh2γ]::Hnl[BT2(aκT1atanhγ)Csinh2γ]:eΩ:.
:Hm(μa+νa):=2m(μa+νa)m,
Ln(xy)=l=0n(1)nln!l![(nl)!]2(xy)nl,
ρsn(t)=n!22nAB2n+1κnT1n(sinh2γ)n:Ln(BT22(aκT1atanhγ)(aκT1atanhγ)κT1)eΩ:,
ρsn(t;g=0)=n!22nAE2n+1(1e2κt)n(sinh2γ)n:Ln(E(aκT1atanhγ)(aκT1atanhγ)1e2κt)×exp[Etanhγ2e2κt(a2+a2)+E[(1e2κt)tanh2γ1]aa]:,
ρn(t)=κnT1n(1gT1):Ln(T22aaκT1)exp[(gT11)aa]:.
eλaaexp[(eλ1)aa]:,
ρn(t)=l=0n(1gT1)n!κlT1lT22(nl)l![(nl)!]2anlexp[aaln(gT1)]anl.
ρn(t;g=0)=l=0n(nl)(1e2κt)nl(e2κt)l|ll|,
ρsv(t)=(1gT1)Bcoshγ:exp[BT22tanhγ2(a2+a2)+(BκT1T22tanh2γ+gT11)aa]:.
Δ(z,z*)=exp(2|z|2)d2zπ2|zz|exp[2(zz*z*z)],
Wsn(z,t)=Tr[ρsn(t)Δ(z,z*)].
Wsn(z,t)=ABC2nπg1l=0nk=0nl(n!)222lBlκlT1ll!k![(nlk)!]2(1g2)nlkg3kC2l(sinh2γ)l×|Hnlk(g421g2)|2exp{4h5g12(z*2+z2)+[24(h6+2)g12]|z|2},
h3=BT2Csinh2γ,h4=BκT1T2tanhγCsinh2γ,h5=BT22tanhγ2,h6=BκT1T22tanh2γ+gT11,g1=[(h6+2)24h52]1/2,g2=4g12[h52(h32+h42)h3h4(h6+2)],g3=4g12[4h3h4h5(h6+2)(h32+h42)],g4=4g12{[(h6+2)h42h3h5]z+[(h6+2)h32h4h5]z*}.
Wn(z,t)=l=0nk=0nln!(1gT1)κlT1l(1)kT22(nl)22(nlk)πl!k![(nlk)!]2(gT1+1)2(nl)k+1×|z|2(nlk)exp{[24gT1+1]|z|2},
Wsv(z,t)=(1gT1)Bπg1coshγexp{4h5g12(z*2+z2)+[24(h6+2)g12]|z|2}.
Wsn(z,0)=n!(tanhγ)nπ22nk=0n23k/2(2+tanhγ)nk/2k![(nk)!]2tanhkγ|Hnk[2(z*+ztanhγ)coshγ(2+tanhγ)tanhγ]|2×exp[(z*2+z2)sinh2γ2|z|2cosh2γ],
ρst=S(γ)ρthS1(γ),
ρth=2(1eλ)eλ+1::exp[eλ1eλ+1(Q2+P2)]::,
ρst=2(1eλ)eλ+1::exp[eλ1eλ+1(e2γQ2+e2γP2)]::.
ρst=2fh:exp[fh2(a2+a2)(f+h)aa]:,
ρst=2fhexp(fh2a2)exp{aaln[1(f+h)]}exp(fh2a2).
ρst=A1exp[A2(a2+a2)+A3aa],
A1=1n¯2(2n¯+1)sinh2γ,A2=(2n¯+1)sinh2γ4[n¯2(2n¯+1)sinh2γ],A3=1(2n¯+1)cosh2γ2[n¯2(2n¯+1)sinh2γ],
ρst(t)=A1(1gT1)i,j=0κigji!j!T1i+jajeaalnT2aiexp[A2(a2+a2)+A3aa]aieaalnT2aj.
ρst(t)=A1(1gT1)F24A22:exp[(G1)aa+H(a2+a2)]:,
ρst(t)=A1(1gT1)F24A22exp(Ha2)exp[aalnG]exp(Ha2),
ρst(t;g=0)=A1Lexp(A2Le2κta2)exp{aaln[L(e2κtA3)e2κt]}exp(A2Le2κta2),
ρth(t)=1gT1n¯(1κT1)+1exp{aaln[n¯T22n¯(1κT1)+1+gT1]}.
Wst(z,t)=A1(1gT1)π(F24A22)[(G+1)24H2]×exp{[4(G+1)(G+1)24H2+2]|z|2+4H(z2+z*2)(G+1)24H2}.
ρsn(t)=Ai,j=0κigji!j!T1i+jaj:d2αd2βπ2αiHn(α*sinh2γ)Hn(βsinh2γ)×exp[|α|2|β|2tanhγ2(α*2+β2)+αT2a+β*T2aaa]:β*iaj.
Hn(q)=nτnexp(2qττ2)|τ=0,
ρsn(t)=A2nτnsn:d2αd2βπ2exp(2α*τsinh2γτ2+2βssinh2γs2)×exp[|α|2|β|2tanhγ2(α*2+β2)+αT2a+β*T2a+κT1αβ*+(gT11)aa]|s=t=0:.
d2zπexp(ζ|z|2+ξz+λz*+fz2+gz*2)=1ζ24fgexp[ζξλ+ξ2g+λ2fζ24fg],
ρsn(t)=AB:2nτnsnexp[DτC2τ2]exp[(D4BκT1τsinh2γ)sC2s2]|s=t=0×exp[BT22tanhγ2(a2+a2)+(BκT1T22tanh2γ+gT11)aa]:,
B=(1κ2T12tanh2γ)1,C=B(1κ2T12),D=2BT2(aκT1atanhγ)/sinh2γ.
Wsn(z,t)=h1exp(2|z|2)l=0nh2d2zπ2z|:Hnl[h3a+h4a]Hnl[h3a+h4a]×exp[h5(a2+a2)+h6aa]:|zexp[2(zz*z*z)]=h1exp(2|z|2)l=0nh2d2zπ2Hnl[h3z*+h4z]Hnl[h3zh4z*]×exp[h5(z*2+z2)(h6+2)|z|2+2(zz*z*z)],
h1=ABC2n,h2=(n!)222lBlκlT1ll![(nl)!]2C2l(sinh2γ)l,h3=BT2Csinh2γ,h4=BκT1T2tanhγCsinh2γ,h5=BT22tanhγ2,h6=BκT1T22tanh2γ+gT11.
Wsn(z,t)=h1exp(2|z|2)l=0nh22(nl)τnlsnld2zπ2×exp[2(h3s+h4τ)z2(h4s+h3τ)z*τ2s2]|s=t=0×exp[(h6+2)|z|2+h5(z*2+z2)+2(zz*z*z)]=h1l=0nh2g1exp{4h5g12(z*2+z2)+[24(h6+2)g12]|z|2}×2(nl)τnlsnlexp[(1g2)(τ2+s2)+g3sτ+g4τ+g4*s]|s=t=0,
g1=[(h6+2)24h52]1/2,g2=4h72[h52(h32+h42)h3h4(h6+2)],g3=4h72[4h3h4h5(h6+2)(h32+h42)],g4=4h72{[(h6+2)h42h3h5]z+[(h6+2)h32h4h5]z*}.

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