Abstract

Extending the recent work completed by Biswas and Agarwal [Phys. Rev. A 75, 032104 (2007)] to the case of m-photon-subtracted squeezed vacuum states (m-PSSVSs), we focus our study on nonclassicality and decoherence of the m-PSSVSs. The nonclassical properties are investigated in terms of the squeezing character, the oscillation of photon-number distribution, and the partial negativity of Wigner function (WF).We then study the effect of decoherence on the m-PSSVSs in two different channels, viz., thermal process and phase damping. In each case, the time-evolution of density operators and WFs of such states is derived analytically. After undergoing the thermal channel, the initial pure m-PSSVSs evolve into a mixed state that turns out to be a Laguerre polynomial of combination of creation and annihilation operators within normal ordering; however, they become another mixed state with the exponential decay due to phase damping. At long times, these fields decay to a highly classical thermal field as a result of thermal noise, but they still keep nonclassicality in the phase-damping channel.

© 2012 Optical Society of America

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  1. J. Eisert, S. Scheel, and M. B. Plenio, “Distilling Gaussian states with Gaussian operations is impossible,” Phys. Rev. Lett. 89, 137903 (2002).
    [CrossRef]
  2. H. Takahashi, J. S. Neergaard-Nielsenl, M. Takeuchil, M. Takeokal, K. Hayasakal, A. Furusawa, and M. Sasakil, “Entanglement distillation from Gaussian input states,” Nat. Photonics 4, 178–181 (2010).
    [CrossRef]
  3. Y.-Q. Zhang and J.-B. Xu, “Entanglement swapping with non-Gaussian resources,” J. Mod. Opt. 58, 593–598 (2011).
    [CrossRef]
  4. S. L. Braunstein and H. J. Kimble, “Teleportation of continuous quantum variables,” Phys. Rev. Lett. 80, 869–872 (1998).
    [CrossRef]
  5. Y. Yang and F.-L. Li, “Entanglement properties of non-Gaussian resources generated via photon subtraction and addition and continuous-variable quantum-teleportation improvement,” Phys. Rev. A 80, 022315 (2009).
    [CrossRef]
  6. N. J. Cerf, O. Krüger, P. Navez, R. F. Werner, and M. M. Wolf, “Non-Gaussian cloning of quantum coherent states is optimal,” Phys. Rev. Lett. 95, 070501 (2005).
    [CrossRef]
  7. P. van Loock, W. J. Munro, K. Nemoto, T. P. Spiller, T. D. Ladd, S. L. Braunstein, and G. J. Milburn, “Hybrid quantum computation in quantum optics,” Phys. Rev. A 78, 022303 (2008).
    [CrossRef]
  8. T. C. Ralph, A. Gilchrist, G. J. Milburn, W. J. Munro, and S. Glancy, “Quantum computation with optical coherent states,” Phys. Rev. A 68, 042319 (2003).
    [CrossRef]
  9. T. Opatrný, G. Kurizki, and D.-G. Welsch, “Improvement on teleportation of continuous variables by photon subtraction via conditional measurement,” Phys. Rev. A 61, 032302 (2000).
    [CrossRef]
  10. M. S. Kim, E. Park, P. L. Knight, and H. Jeong, “Nonclassicality of a photon-subtracted Gaussian field,” Phys. Rev. A 71, 043805 (2005).
    [CrossRef]
  11. A. Kitagawa, M. Takeoka, M. Sasaki, and A. Chefles, “Entanglement evaluation of non-Gaussian states generated by photon subtraction from squeezed states,” Phys. Rev. A 73, 042310 (2006).
    [CrossRef]
  12. L.-y. Hu, X.-x. Xu, Z.-s. Wang, and X.-f. Xu, “Photon-subtracted squeezed thermal state: nonclassicality and decoherence,” Phys. Rev. A 82, 043842 (2010).
    [CrossRef]
  13. S. Olivares and M. G. A. Paris, “Squeezed Fock state by inconclusive photon subtraction,” J. Opt. B: Quantum Semiclass. Opt. 7, S616–S621 (2010).
    [CrossRef]
  14. A. Biswas and G. S. Agarwal, “Nonclassicality and decoherence of photon-subtracted squeezed states,” Phys. Rev. A 75, 032104 (2007).
    [CrossRef]
  15. 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]
  16. J. S. Neergaard-Nielsen, M. Takeuchi, K. Wakui, H. Takahashi, K. Hayasaka, M. Takeoka, and M. Sasaki, “Photon subtraction from traveling fields-recent experimental demonstrations,” Prog. Inform. 8, 5–18 (2011).
    [CrossRef]
  17. K. Wakui, H. Takahashi, A. Furusawa, and M. Sasaki, “Photon subtracted squeezed states generated with periodically poled KTiOPO4,” Opt. Express 15, 3568–3574 (2007).
    [CrossRef]
  18. J. Wenger, R. Tualle-Brouri, and P. Grangier, “Non-Gaussian statistics from individual pulses of squeezed light,” Phys. Rev. Lett. 92, 153601 (2004).
    [CrossRef]
  19. C. Invernizzi, S. Olivares, M. G. A. Paris, and K. Banaszek, “Effect of noise and enhancement of nonlocality in on/off photodetection,” Phys. Rev. A 72, 042105 (2005).
    [CrossRef]
  20. M. Sasaki and S. Suzuki, “Multimode theory of measurement-induced non-Gaussian operation on wideband squeezed light: analytical formula,” Phys. Rev. A 73, 043807 (2006).
    [CrossRef]
  21. F. Dell’Anno, S. De Siena, L. Albano, and F. Illuminati, “Continuous-variable quantum teleportation with non-Gaussian resources,” Phys. Rev. A 76, 022301 (2007).
    [CrossRef]
  22. M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University, 2004).
  23. V. V. Dodonov, “‘Nonclassical’ states in quantum optics: a ‘squeezed’ review of the first 75 years,” J. Opt. B: Quantum Semiclass. Opt. 4, R1–R33 (2002).
    [CrossRef]
  24. H. Y. Fan, “New antinormal ordering expansion for density operators,” Phys. Lett. A 161, 1–4 (1991).
    [CrossRef]
  25. R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131, 2766–2788 (1963).
    [CrossRef]
  26. J. R. Klauder and B. S. Skargerstam, Coherent States (World Scientific, 1985).
  27. W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer, 1996).
  28. R. R. Puri, Mathematical Methods of Quantum Optics (Springer-Verlag, 2001).
  29. G. S. Agarwal and E. Wolf, “Calculus for functions of noncommuting operators and general phase-space methods in quantum mechanics. I. mapping theorems and ordering of functions of noncommuting operators,” Phys. Rev. D 2, 2161–2186(1970).
    [CrossRef]
  30. H. Y. Fan, X. G. Meng, and J. S. Wang, “New form of Legendre polynomials obtained by virtue of excited squeezed state and IWOP technique in quantum optics,” Commun. Theor. Phys. 46, 845–848 (2006).
    [CrossRef]
  31. W. Schleich, D. F. Walls, and J. A. Wheeler, “Area of overlap and interference in phase space versus Wigner pseudoprobabilities,” Phys. Rev. A 38, 1177–1186 (1988).
    [CrossRef]
  32. W. Schleich and J. A. Wheeler, “Oscillations in photon distribution of squeezed states and interference in phase space,” Nature 326, 574–577 (1987).
    [CrossRef]
  33. W. Schleich and J. A. Wheeler, “Oscillations in photon distribution of squeezed states,” J. Opt. Soc. Am. B 4, 1715–1722(1987).
    [CrossRef]
  34. W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, 1973).
  35. C. L. Methta, “Diagonal coherent-state representation of quantum operators,” Phys. Rev. Lett. 18, 752–754 (1967).
    [CrossRef]
  36. E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
    [CrossRef]
  37. P. Schleich Wolfgang, Quantum Optics in Phase Space(Wiley-VCH, 2001).
  38. H. Y. Fan and H. L. Cheng, “Two-parameter Radon transformation of the Wigner operator and its inverse,” Chin. Phys. Lett. 18, 850–853 (2001).
    [CrossRef]
  39. C. W. Gardiner and P. Zoller, Quantum Noise (Springer, 2000).
  40. H. Y. Fan and L. Y. Hu, “Operator-sum representation of density operators as solutions to master equations obtained via the entangled state approach,” Mod. Phys. Lett. B 22, 2435–2468 (2008).
    [CrossRef]

2011 (2)

Y.-Q. Zhang and J.-B. Xu, “Entanglement swapping with non-Gaussian resources,” J. Mod. Opt. 58, 593–598 (2011).
[CrossRef]

J. S. Neergaard-Nielsen, M. Takeuchi, K. Wakui, H. Takahashi, K. Hayasaka, M. Takeoka, and M. Sasaki, “Photon subtraction from traveling fields-recent experimental demonstrations,” Prog. Inform. 8, 5–18 (2011).
[CrossRef]

2010 (3)

H. Takahashi, J. S. Neergaard-Nielsenl, M. Takeuchil, M. Takeokal, K. Hayasakal, A. Furusawa, and M. Sasakil, “Entanglement distillation from Gaussian input states,” Nat. Photonics 4, 178–181 (2010).
[CrossRef]

L.-y. Hu, X.-x. Xu, Z.-s. Wang, and X.-f. Xu, “Photon-subtracted squeezed thermal state: nonclassicality and decoherence,” Phys. Rev. A 82, 043842 (2010).
[CrossRef]

S. Olivares and M. G. A. Paris, “Squeezed Fock state by inconclusive photon subtraction,” J. Opt. B: Quantum Semiclass. Opt. 7, S616–S621 (2010).
[CrossRef]

2009 (1)

Y. Yang and F.-L. Li, “Entanglement properties of non-Gaussian resources generated via photon subtraction and addition and continuous-variable quantum-teleportation improvement,” Phys. Rev. A 80, 022315 (2009).
[CrossRef]

2008 (3)

P. van Loock, W. J. Munro, K. Nemoto, T. P. Spiller, T. D. Ladd, S. L. Braunstein, and G. J. Milburn, “Hybrid quantum computation in quantum optics,” Phys. Rev. A 78, 022303 (2008).
[CrossRef]

H. Y. Fan and L. Y. Hu, “Operator-sum representation of density operators as solutions to master equations obtained via the entangled state 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]

2007 (3)

K. Wakui, H. Takahashi, A. Furusawa, and M. Sasaki, “Photon subtracted squeezed states generated with periodically poled KTiOPO4,” Opt. Express 15, 3568–3574 (2007).
[CrossRef]

F. Dell’Anno, S. De Siena, L. Albano, and F. Illuminati, “Continuous-variable quantum teleportation with non-Gaussian resources,” Phys. Rev. A 76, 022301 (2007).
[CrossRef]

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

2006 (3)

A. Kitagawa, M. Takeoka, M. Sasaki, and A. Chefles, “Entanglement evaluation of non-Gaussian states generated by photon subtraction from squeezed states,” Phys. Rev. A 73, 042310 (2006).
[CrossRef]

M. Sasaki and S. Suzuki, “Multimode theory of measurement-induced non-Gaussian operation on wideband squeezed light: analytical formula,” Phys. Rev. A 73, 043807 (2006).
[CrossRef]

H. Y. Fan, X. G. Meng, and J. S. Wang, “New form of Legendre polynomials obtained by virtue of excited squeezed state and IWOP technique in quantum optics,” Commun. Theor. Phys. 46, 845–848 (2006).
[CrossRef]

2005 (3)

C. Invernizzi, S. Olivares, M. G. A. Paris, and K. Banaszek, “Effect of noise and enhancement of nonlocality in on/off photodetection,” Phys. Rev. A 72, 042105 (2005).
[CrossRef]

M. S. Kim, E. Park, P. L. Knight, and H. Jeong, “Nonclassicality of a photon-subtracted Gaussian field,” Phys. Rev. A 71, 043805 (2005).
[CrossRef]

N. J. Cerf, O. Krüger, P. Navez, R. F. Werner, and M. M. Wolf, “Non-Gaussian cloning of quantum coherent states is optimal,” Phys. Rev. Lett. 95, 070501 (2005).
[CrossRef]

2004 (1)

J. Wenger, R. Tualle-Brouri, and P. Grangier, “Non-Gaussian statistics from individual pulses of squeezed light,” Phys. Rev. Lett. 92, 153601 (2004).
[CrossRef]

2003 (1)

T. C. Ralph, A. Gilchrist, G. J. Milburn, W. J. Munro, and S. Glancy, “Quantum computation with optical coherent states,” Phys. Rev. A 68, 042319 (2003).
[CrossRef]

2002 (2)

J. Eisert, S. Scheel, and M. B. Plenio, “Distilling Gaussian states with Gaussian operations is impossible,” Phys. Rev. Lett. 89, 137903 (2002).
[CrossRef]

V. V. Dodonov, “‘Nonclassical’ states in quantum optics: a ‘squeezed’ review of the first 75 years,” J. Opt. B: Quantum Semiclass. Opt. 4, R1–R33 (2002).
[CrossRef]

2001 (1)

H. Y. Fan and H. L. Cheng, “Two-parameter Radon transformation of the Wigner operator and its inverse,” Chin. Phys. Lett. 18, 850–853 (2001).
[CrossRef]

2000 (1)

T. Opatrný, G. Kurizki, and D.-G. Welsch, “Improvement on teleportation of continuous variables by photon subtraction via conditional measurement,” Phys. Rev. A 61, 032302 (2000).
[CrossRef]

1998 (1)

S. L. Braunstein and H. J. Kimble, “Teleportation of continuous quantum variables,” Phys. Rev. Lett. 80, 869–872 (1998).
[CrossRef]

1991 (1)

H. Y. Fan, “New antinormal ordering expansion for density operators,” Phys. Lett. A 161, 1–4 (1991).
[CrossRef]

1988 (1)

W. Schleich, D. F. Walls, and J. A. Wheeler, “Area of overlap and interference in phase space versus Wigner pseudoprobabilities,” Phys. Rev. A 38, 1177–1186 (1988).
[CrossRef]

1987 (2)

W. Schleich and J. A. Wheeler, “Oscillations in photon distribution of squeezed states and interference in phase space,” Nature 326, 574–577 (1987).
[CrossRef]

W. Schleich and J. A. Wheeler, “Oscillations in photon distribution of squeezed states,” J. Opt. Soc. Am. B 4, 1715–1722(1987).
[CrossRef]

1970 (1)

G. S. Agarwal and E. Wolf, “Calculus for functions of noncommuting operators and general phase-space methods in quantum mechanics. I. mapping theorems and ordering of functions of noncommuting operators,” Phys. Rev. D 2, 2161–2186(1970).
[CrossRef]

1967 (1)

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

1963 (1)

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

1932 (1)

E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[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]

G. S. Agarwal and E. Wolf, “Calculus for functions of noncommuting operators and general phase-space methods in quantum mechanics. I. mapping theorems and ordering of functions of noncommuting operators,” Phys. Rev. D 2, 2161–2186(1970).
[CrossRef]

Albano, L.

F. Dell’Anno, S. De Siena, L. Albano, and F. Illuminati, “Continuous-variable quantum teleportation with non-Gaussian resources,” Phys. Rev. A 76, 022301 (2007).
[CrossRef]

Banaszek, K.

C. Invernizzi, S. Olivares, M. G. A. Paris, and K. Banaszek, “Effect of noise and enhancement of nonlocality in on/off photodetection,” Phys. Rev. A 72, 042105 (2005).
[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.

P. van Loock, W. J. Munro, K. Nemoto, T. P. Spiller, T. D. Ladd, S. L. Braunstein, and G. J. Milburn, “Hybrid quantum computation in quantum optics,” Phys. Rev. A 78, 022303 (2008).
[CrossRef]

S. L. Braunstein and H. J. Kimble, “Teleportation of continuous quantum variables,” Phys. Rev. Lett. 80, 869–872 (1998).
[CrossRef]

Cerf, N. J.

N. J. Cerf, O. Krüger, P. Navez, R. F. Werner, and M. M. Wolf, “Non-Gaussian cloning of quantum coherent states is optimal,” Phys. Rev. Lett. 95, 070501 (2005).
[CrossRef]

Chefles, A.

A. Kitagawa, M. Takeoka, M. Sasaki, and A. Chefles, “Entanglement evaluation of non-Gaussian states generated by photon subtraction from squeezed states,” Phys. Rev. A 73, 042310 (2006).
[CrossRef]

Cheng, H. L.

H. Y. Fan and H. L. Cheng, “Two-parameter Radon transformation of the Wigner operator and its inverse,” Chin. Phys. Lett. 18, 850–853 (2001).
[CrossRef]

De Siena, S.

F. Dell’Anno, S. De Siena, L. Albano, and F. Illuminati, “Continuous-variable quantum teleportation with non-Gaussian resources,” Phys. Rev. A 76, 022301 (2007).
[CrossRef]

Dell’Anno, F.

F. Dell’Anno, S. De Siena, L. Albano, and F. Illuminati, “Continuous-variable quantum teleportation with non-Gaussian resources,” Phys. Rev. A 76, 022301 (2007).
[CrossRef]

Dodonov, V. V.

V. V. Dodonov, “‘Nonclassical’ states in quantum optics: a ‘squeezed’ review of the first 75 years,” J. Opt. B: Quantum Semiclass. Opt. 4, R1–R33 (2002).
[CrossRef]

Eisert, J.

J. Eisert, S. Scheel, and M. B. Plenio, “Distilling Gaussian states with Gaussian operations is impossible,” Phys. Rev. Lett. 89, 137903 (2002).
[CrossRef]

Fan, H. Y.

H. Y. Fan and L. Y. Hu, “Operator-sum representation of density operators as solutions to master equations obtained via the entangled state 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, X. G. Meng, and J. S. Wang, “New form of Legendre polynomials obtained by virtue of excited squeezed state and IWOP technique in quantum optics,” Commun. Theor. Phys. 46, 845–848 (2006).
[CrossRef]

H. Y. Fan and H. L. Cheng, “Two-parameter Radon transformation of the Wigner operator and its inverse,” Chin. Phys. Lett. 18, 850–853 (2001).
[CrossRef]

H. Y. Fan, “New antinormal ordering expansion for density operators,” Phys. Lett. A 161, 1–4 (1991).
[CrossRef]

Furusawa, A.

H. Takahashi, J. S. Neergaard-Nielsenl, M. Takeuchil, M. Takeokal, K. Hayasakal, A. Furusawa, and M. Sasakil, “Entanglement distillation from Gaussian input states,” Nat. Photonics 4, 178–181 (2010).
[CrossRef]

K. Wakui, H. Takahashi, A. Furusawa, and M. Sasaki, “Photon subtracted squeezed states generated with periodically poled KTiOPO4,” Opt. Express 15, 3568–3574 (2007).
[CrossRef]

Gardiner, C. W.

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

Gilchrist, A.

T. C. Ralph, A. Gilchrist, G. J. Milburn, W. J. Munro, and S. Glancy, “Quantum computation with optical coherent states,” Phys. Rev. A 68, 042319 (2003).
[CrossRef]

Glancy, S.

T. C. Ralph, A. Gilchrist, G. J. Milburn, W. J. Munro, and S. Glancy, “Quantum computation with optical coherent states,” Phys. Rev. A 68, 042319 (2003).
[CrossRef]

Glauber, R. J.

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

Grangier, P.

J. Wenger, R. Tualle-Brouri, and P. Grangier, “Non-Gaussian statistics from individual pulses of squeezed light,” Phys. Rev. Lett. 92, 153601 (2004).
[CrossRef]

Hayasaka, K.

J. S. Neergaard-Nielsen, M. Takeuchi, K. Wakui, H. Takahashi, K. Hayasaka, M. Takeoka, and M. Sasaki, “Photon subtraction from traveling fields-recent experimental demonstrations,” Prog. Inform. 8, 5–18 (2011).
[CrossRef]

Hayasakal, K.

H. Takahashi, J. S. Neergaard-Nielsenl, M. Takeuchil, M. Takeokal, K. Hayasakal, A. Furusawa, and M. Sasakil, “Entanglement distillation from Gaussian input states,” Nat. Photonics 4, 178–181 (2010).
[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 solutions to master equations obtained via the entangled state approach,” Mod. Phys. Lett. B 22, 2435–2468 (2008).
[CrossRef]

Hu, L.-y.

L.-y. Hu, X.-x. Xu, Z.-s. Wang, and X.-f. Xu, “Photon-subtracted squeezed thermal state: nonclassicality and decoherence,” Phys. Rev. A 82, 043842 (2010).
[CrossRef]

Illuminati, F.

F. Dell’Anno, S. De Siena, L. Albano, and F. Illuminati, “Continuous-variable quantum teleportation with non-Gaussian resources,” Phys. Rev. A 76, 022301 (2007).
[CrossRef]

Invernizzi, C.

C. Invernizzi, S. Olivares, M. G. A. Paris, and K. Banaszek, “Effect of noise and enhancement of nonlocality in on/off photodetection,” Phys. Rev. A 72, 042105 (2005).
[CrossRef]

Jeong, H.

M. S. Kim, E. Park, P. L. Knight, and H. Jeong, “Nonclassicality of a photon-subtracted Gaussian field,” Phys. Rev. A 71, 043805 (2005).
[CrossRef]

Kim, M. S.

M. S. Kim, E. Park, P. L. Knight, and H. Jeong, “Nonclassicality of a photon-subtracted Gaussian field,” Phys. Rev. A 71, 043805 (2005).
[CrossRef]

Kimble, H. J.

S. L. Braunstein and H. J. Kimble, “Teleportation of continuous quantum variables,” Phys. Rev. Lett. 80, 869–872 (1998).
[CrossRef]

Kitagawa, A.

A. Kitagawa, M. Takeoka, M. Sasaki, and A. Chefles, “Entanglement evaluation of non-Gaussian states generated by photon subtraction from squeezed states,” Phys. Rev. A 73, 042310 (2006).
[CrossRef]

Klauder, J. R.

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

Knight, P. L.

M. S. Kim, E. Park, P. L. Knight, and H. Jeong, “Nonclassicality of a photon-subtracted Gaussian field,” Phys. Rev. A 71, 043805 (2005).
[CrossRef]

Krüger, O.

N. J. Cerf, O. Krüger, P. Navez, R. F. Werner, and M. M. Wolf, “Non-Gaussian cloning of quantum coherent states is optimal,” Phys. Rev. Lett. 95, 070501 (2005).
[CrossRef]

Kurizki, G.

T. Opatrný, G. Kurizki, and D.-G. Welsch, “Improvement on teleportation of continuous variables by photon subtraction via conditional measurement,” Phys. Rev. A 61, 032302 (2000).
[CrossRef]

Ladd, T. D.

P. van Loock, W. J. Munro, K. Nemoto, T. P. Spiller, T. D. Ladd, S. L. Braunstein, and G. J. Milburn, “Hybrid quantum computation in quantum optics,” Phys. Rev. A 78, 022303 (2008).
[CrossRef]

Li, F.-L.

Y. Yang and F.-L. Li, “Entanglement properties of non-Gaussian resources generated via photon subtraction and addition and continuous-variable quantum-teleportation improvement,” Phys. Rev. A 80, 022315 (2009).
[CrossRef]

Louisell, W. H.

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

Magnus, W.

W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer, 1996).

Meng, X. G.

H. Y. Fan, X. G. Meng, and J. S. Wang, “New form of Legendre polynomials obtained by virtue of excited squeezed state and IWOP technique in quantum optics,” Commun. Theor. Phys. 46, 845–848 (2006).
[CrossRef]

Methta, C. L.

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

Milburn, G. J.

P. van Loock, W. J. Munro, K. Nemoto, T. P. Spiller, T. D. Ladd, S. L. Braunstein, and G. J. Milburn, “Hybrid quantum computation in quantum optics,” Phys. Rev. A 78, 022303 (2008).
[CrossRef]

T. C. Ralph, A. Gilchrist, G. J. Milburn, W. J. Munro, and S. Glancy, “Quantum computation with optical coherent states,” Phys. Rev. A 68, 042319 (2003).
[CrossRef]

Munro, W. J.

P. van Loock, W. J. Munro, K. Nemoto, T. P. Spiller, T. D. Ladd, S. L. Braunstein, and G. J. Milburn, “Hybrid quantum computation in quantum optics,” Phys. Rev. A 78, 022303 (2008).
[CrossRef]

T. C. Ralph, A. Gilchrist, G. J. Milburn, W. J. Munro, and S. Glancy, “Quantum computation with optical coherent states,” Phys. Rev. A 68, 042319 (2003).
[CrossRef]

Navez, P.

N. J. Cerf, O. Krüger, P. Navez, R. F. Werner, and M. M. Wolf, “Non-Gaussian cloning of quantum coherent states is optimal,” Phys. Rev. Lett. 95, 070501 (2005).
[CrossRef]

Neergaard-Nielsen, J. S.

J. S. Neergaard-Nielsen, M. Takeuchi, K. Wakui, H. Takahashi, K. Hayasaka, M. Takeoka, and M. Sasaki, “Photon subtraction from traveling fields-recent experimental demonstrations,” Prog. Inform. 8, 5–18 (2011).
[CrossRef]

Neergaard-Nielsenl, J. S.

H. Takahashi, J. S. Neergaard-Nielsenl, M. Takeuchil, M. Takeokal, K. Hayasakal, A. Furusawa, and M. Sasakil, “Entanglement distillation from Gaussian input states,” Nat. Photonics 4, 178–181 (2010).
[CrossRef]

Nemoto, K.

P. van Loock, W. J. Munro, K. Nemoto, T. P. Spiller, T. D. Ladd, S. L. Braunstein, and G. J. Milburn, “Hybrid quantum computation in quantum optics,” Phys. Rev. A 78, 022303 (2008).
[CrossRef]

Oberhettinger, F.

W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer, 1996).

Olivares, S.

S. Olivares and M. G. A. Paris, “Squeezed Fock state by inconclusive photon subtraction,” J. Opt. B: Quantum Semiclass. Opt. 7, S616–S621 (2010).
[CrossRef]

C. Invernizzi, S. Olivares, M. G. A. Paris, and K. Banaszek, “Effect of noise and enhancement of nonlocality in on/off photodetection,” Phys. Rev. A 72, 042105 (2005).
[CrossRef]

Opatrný, T.

T. Opatrný, G. Kurizki, and D.-G. Welsch, “Improvement on teleportation of continuous variables by photon subtraction via conditional measurement,” Phys. Rev. A 61, 032302 (2000).
[CrossRef]

Paris, M. G. A.

S. Olivares and M. G. A. Paris, “Squeezed Fock state by inconclusive photon subtraction,” J. Opt. B: Quantum Semiclass. Opt. 7, S616–S621 (2010).
[CrossRef]

C. Invernizzi, S. Olivares, M. G. A. Paris, and K. Banaszek, “Effect of noise and enhancement of nonlocality in on/off photodetection,” Phys. Rev. A 72, 042105 (2005).
[CrossRef]

Park, E.

M. S. Kim, E. Park, P. L. Knight, and H. Jeong, “Nonclassicality of a photon-subtracted Gaussian field,” Phys. Rev. A 71, 043805 (2005).
[CrossRef]

Plenio, M. B.

J. Eisert, S. Scheel, and M. B. Plenio, “Distilling Gaussian states with Gaussian operations is impossible,” Phys. Rev. Lett. 89, 137903 (2002).
[CrossRef]

Puri, R. R.

R. R. Puri, Mathematical Methods of Quantum Optics (Springer-Verlag, 2001).

Ralph, T. C.

T. C. Ralph, A. Gilchrist, G. J. Milburn, W. J. Munro, and S. Glancy, “Quantum computation with optical coherent states,” Phys. Rev. A 68, 042319 (2003).
[CrossRef]

Sasaki, M.

J. S. Neergaard-Nielsen, M. Takeuchi, K. Wakui, H. Takahashi, K. Hayasaka, M. Takeoka, and M. Sasaki, “Photon subtraction from traveling fields-recent experimental demonstrations,” Prog. Inform. 8, 5–18 (2011).
[CrossRef]

K. Wakui, H. Takahashi, A. Furusawa, and M. Sasaki, “Photon subtracted squeezed states generated with periodically poled KTiOPO4,” Opt. Express 15, 3568–3574 (2007).
[CrossRef]

A. Kitagawa, M. Takeoka, M. Sasaki, and A. Chefles, “Entanglement evaluation of non-Gaussian states generated by photon subtraction from squeezed states,” Phys. Rev. A 73, 042310 (2006).
[CrossRef]

M. Sasaki and S. Suzuki, “Multimode theory of measurement-induced non-Gaussian operation on wideband squeezed light: analytical formula,” Phys. Rev. A 73, 043807 (2006).
[CrossRef]

Sasakil, M.

H. Takahashi, J. S. Neergaard-Nielsenl, M. Takeuchil, M. Takeokal, K. Hayasakal, A. Furusawa, and M. Sasakil, “Entanglement distillation from Gaussian input states,” Nat. Photonics 4, 178–181 (2010).
[CrossRef]

Scheel, S.

J. Eisert, S. Scheel, and M. B. Plenio, “Distilling Gaussian states with Gaussian operations is impossible,” Phys. Rev. Lett. 89, 137903 (2002).
[CrossRef]

Schleich, W.

W. Schleich, D. F. Walls, and J. A. Wheeler, “Area of overlap and interference in phase space versus Wigner pseudoprobabilities,” Phys. Rev. A 38, 1177–1186 (1988).
[CrossRef]

W. Schleich and J. A. Wheeler, “Oscillations in photon distribution of squeezed states,” J. Opt. Soc. Am. B 4, 1715–1722(1987).
[CrossRef]

W. Schleich and J. A. Wheeler, “Oscillations in photon distribution of squeezed states and interference in phase space,” Nature 326, 574–577 (1987).
[CrossRef]

Scully, M. O.

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

Skargerstam, B. S.

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

Soni, R. P.

W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer, 1996).

Spiller, T. P.

P. van Loock, W. J. Munro, K. Nemoto, T. P. Spiller, T. D. Ladd, S. L. Braunstein, and G. J. Milburn, “Hybrid quantum computation in quantum optics,” Phys. Rev. A 78, 022303 (2008).
[CrossRef]

Suzuki, S.

M. Sasaki and S. Suzuki, “Multimode theory of measurement-induced non-Gaussian operation on wideband squeezed light: analytical formula,” Phys. Rev. A 73, 043807 (2006).
[CrossRef]

Takahashi, H.

J. S. Neergaard-Nielsen, M. Takeuchi, K. Wakui, H. Takahashi, K. Hayasaka, M. Takeoka, and M. Sasaki, “Photon subtraction from traveling fields-recent experimental demonstrations,” Prog. Inform. 8, 5–18 (2011).
[CrossRef]

H. Takahashi, J. S. Neergaard-Nielsenl, M. Takeuchil, M. Takeokal, K. Hayasakal, A. Furusawa, and M. Sasakil, “Entanglement distillation from Gaussian input states,” Nat. Photonics 4, 178–181 (2010).
[CrossRef]

K. Wakui, H. Takahashi, A. Furusawa, and M. Sasaki, “Photon subtracted squeezed states generated with periodically poled KTiOPO4,” Opt. Express 15, 3568–3574 (2007).
[CrossRef]

Takeoka, M.

J. S. Neergaard-Nielsen, M. Takeuchi, K. Wakui, H. Takahashi, K. Hayasaka, M. Takeoka, and M. Sasaki, “Photon subtraction from traveling fields-recent experimental demonstrations,” Prog. Inform. 8, 5–18 (2011).
[CrossRef]

A. Kitagawa, M. Takeoka, M. Sasaki, and A. Chefles, “Entanglement evaluation of non-Gaussian states generated by photon subtraction from squeezed states,” Phys. Rev. A 73, 042310 (2006).
[CrossRef]

Takeokal, M.

H. Takahashi, J. S. Neergaard-Nielsenl, M. Takeuchil, M. Takeokal, K. Hayasakal, A. Furusawa, and M. Sasakil, “Entanglement distillation from Gaussian input states,” Nat. Photonics 4, 178–181 (2010).
[CrossRef]

Takeuchi, M.

J. S. Neergaard-Nielsen, M. Takeuchi, K. Wakui, H. Takahashi, K. Hayasaka, M. Takeoka, and M. Sasaki, “Photon subtraction from traveling fields-recent experimental demonstrations,” Prog. Inform. 8, 5–18 (2011).
[CrossRef]

Takeuchil, M.

H. Takahashi, J. S. Neergaard-Nielsenl, M. Takeuchil, M. Takeokal, K. Hayasakal, A. Furusawa, and M. Sasakil, “Entanglement distillation from Gaussian input states,” Nat. Photonics 4, 178–181 (2010).
[CrossRef]

Tualle-Brouri, R.

J. Wenger, R. Tualle-Brouri, and P. Grangier, “Non-Gaussian statistics from individual pulses of squeezed light,” Phys. Rev. Lett. 92, 153601 (2004).
[CrossRef]

van Loock, P.

P. van Loock, W. J. Munro, K. Nemoto, T. P. Spiller, T. D. Ladd, S. L. Braunstein, and G. J. Milburn, “Hybrid quantum computation in quantum optics,” Phys. Rev. A 78, 022303 (2008).
[CrossRef]

Wakui, K.

J. S. Neergaard-Nielsen, M. Takeuchi, K. Wakui, H. Takahashi, K. Hayasaka, M. Takeoka, and M. Sasaki, “Photon subtraction from traveling fields-recent experimental demonstrations,” Prog. Inform. 8, 5–18 (2011).
[CrossRef]

K. Wakui, H. Takahashi, A. Furusawa, and M. Sasaki, “Photon subtracted squeezed states generated with periodically poled KTiOPO4,” Opt. Express 15, 3568–3574 (2007).
[CrossRef]

Walls, D. F.

W. Schleich, D. F. Walls, and J. A. Wheeler, “Area of overlap and interference in phase space versus Wigner pseudoprobabilities,” Phys. Rev. A 38, 1177–1186 (1988).
[CrossRef]

Wang, J. S.

H. Y. Fan, X. G. Meng, and J. S. Wang, “New form of Legendre polynomials obtained by virtue of excited squeezed state and IWOP technique in quantum optics,” Commun. Theor. Phys. 46, 845–848 (2006).
[CrossRef]

Wang, Z.-s.

L.-y. Hu, X.-x. Xu, Z.-s. Wang, and X.-f. Xu, “Photon-subtracted squeezed thermal state: nonclassicality and decoherence,” Phys. Rev. A 82, 043842 (2010).
[CrossRef]

Welsch, D.-G.

T. Opatrný, G. Kurizki, and D.-G. Welsch, “Improvement on teleportation of continuous variables by photon subtraction via conditional measurement,” Phys. Rev. A 61, 032302 (2000).
[CrossRef]

Wenger, J.

J. Wenger, R. Tualle-Brouri, and P. Grangier, “Non-Gaussian statistics from individual pulses of squeezed light,” Phys. Rev. Lett. 92, 153601 (2004).
[CrossRef]

Werner, R. F.

N. J. Cerf, O. Krüger, P. Navez, R. F. Werner, and M. M. Wolf, “Non-Gaussian cloning of quantum coherent states is optimal,” Phys. Rev. Lett. 95, 070501 (2005).
[CrossRef]

Wheeler, J. A.

W. Schleich, D. F. Walls, and J. A. Wheeler, “Area of overlap and interference in phase space versus Wigner pseudoprobabilities,” Phys. Rev. A 38, 1177–1186 (1988).
[CrossRef]

W. Schleich and J. A. Wheeler, “Oscillations in photon distribution of squeezed states,” J. Opt. Soc. Am. B 4, 1715–1722(1987).
[CrossRef]

W. Schleich and J. A. Wheeler, “Oscillations in photon distribution of squeezed states and interference in phase space,” Nature 326, 574–577 (1987).
[CrossRef]

Wigner, E. P.

E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Wolf, E.

G. S. Agarwal and E. Wolf, “Calculus for functions of noncommuting operators and general phase-space methods in quantum mechanics. I. mapping theorems and ordering of functions of noncommuting operators,” Phys. Rev. D 2, 2161–2186(1970).
[CrossRef]

Wolf, M. M.

N. J. Cerf, O. Krüger, P. Navez, R. F. Werner, and M. M. Wolf, “Non-Gaussian cloning of quantum coherent states is optimal,” Phys. Rev. Lett. 95, 070501 (2005).
[CrossRef]

Wolfgang, P. Schleich

P. Schleich Wolfgang, Quantum Optics in Phase Space(Wiley-VCH, 2001).

Xu, J.-B.

Y.-Q. Zhang and J.-B. Xu, “Entanglement swapping with non-Gaussian resources,” J. Mod. Opt. 58, 593–598 (2011).
[CrossRef]

Xu, X.-f.

L.-y. Hu, X.-x. Xu, Z.-s. Wang, and X.-f. Xu, “Photon-subtracted squeezed thermal state: nonclassicality and decoherence,” Phys. Rev. A 82, 043842 (2010).
[CrossRef]

Xu, X.-x.

L.-y. Hu, X.-x. Xu, Z.-s. Wang, and X.-f. Xu, “Photon-subtracted squeezed thermal state: nonclassicality and decoherence,” Phys. Rev. A 82, 043842 (2010).
[CrossRef]

Yang, Y.

Y. Yang and F.-L. Li, “Entanglement properties of non-Gaussian resources generated via photon subtraction and addition and continuous-variable quantum-teleportation improvement,” Phys. Rev. A 80, 022315 (2009).
[CrossRef]

Zhang, Y.-Q.

Y.-Q. Zhang and J.-B. Xu, “Entanglement swapping with non-Gaussian resources,” J. Mod. Opt. 58, 593–598 (2011).
[CrossRef]

Zoller, P.

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

Zubairy, M. S.

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

Chin. Phys. Lett. (1)

H. Y. Fan and H. L. Cheng, “Two-parameter Radon transformation of the Wigner operator and its inverse,” Chin. Phys. Lett. 18, 850–853 (2001).
[CrossRef]

Commun. Theor. Phys. (1)

H. Y. Fan, X. G. Meng, and J. S. Wang, “New form of Legendre polynomials obtained by virtue of excited squeezed state and IWOP technique in quantum optics,” Commun. Theor. Phys. 46, 845–848 (2006).
[CrossRef]

J. Mod. Opt. (1)

Y.-Q. Zhang and J.-B. Xu, “Entanglement swapping with non-Gaussian resources,” J. Mod. Opt. 58, 593–598 (2011).
[CrossRef]

J. Opt. B: Quantum Semiclass. Opt. (2)

V. V. Dodonov, “‘Nonclassical’ states in quantum optics: a ‘squeezed’ review of the first 75 years,” J. Opt. B: Quantum Semiclass. Opt. 4, R1–R33 (2002).
[CrossRef]

S. Olivares and M. G. A. Paris, “Squeezed Fock state by inconclusive photon subtraction,” J. Opt. B: Quantum Semiclass. Opt. 7, S616–S621 (2010).
[CrossRef]

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

Mod. Phys. Lett. B (1)

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

Nat. Photonics (1)

H. Takahashi, J. S. Neergaard-Nielsenl, M. Takeuchil, M. Takeokal, K. Hayasakal, A. Furusawa, and M. Sasakil, “Entanglement distillation from Gaussian input states,” Nat. Photonics 4, 178–181 (2010).
[CrossRef]

Nature (1)

W. Schleich and J. A. Wheeler, “Oscillations in photon distribution of squeezed states and interference in phase space,” Nature 326, 574–577 (1987).
[CrossRef]

Opt. Express (1)

Phys. Lett. A (1)

H. Y. Fan, “New antinormal ordering expansion for density operators,” Phys. Lett. A 161, 1–4 (1991).
[CrossRef]

Phys. Rev. (2)

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

E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Phys. Rev. A (12)

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

Y. Yang and F.-L. Li, “Entanglement properties of non-Gaussian resources generated via photon subtraction and addition and continuous-variable quantum-teleportation improvement,” Phys. Rev. A 80, 022315 (2009).
[CrossRef]

P. van Loock, W. J. Munro, K. Nemoto, T. P. Spiller, T. D. Ladd, S. L. Braunstein, and G. J. Milburn, “Hybrid quantum computation in quantum optics,” Phys. Rev. A 78, 022303 (2008).
[CrossRef]

T. C. Ralph, A. Gilchrist, G. J. Milburn, W. J. Munro, and S. Glancy, “Quantum computation with optical coherent states,” Phys. Rev. A 68, 042319 (2003).
[CrossRef]

T. Opatrný, G. Kurizki, and D.-G. Welsch, “Improvement on teleportation of continuous variables by photon subtraction via conditional measurement,” Phys. Rev. A 61, 032302 (2000).
[CrossRef]

M. S. Kim, E. Park, P. L. Knight, and H. Jeong, “Nonclassicality of a photon-subtracted Gaussian field,” Phys. Rev. A 71, 043805 (2005).
[CrossRef]

A. Kitagawa, M. Takeoka, M. Sasaki, and A. Chefles, “Entanglement evaluation of non-Gaussian states generated by photon subtraction from squeezed states,” Phys. Rev. A 73, 042310 (2006).
[CrossRef]

L.-y. Hu, X.-x. Xu, Z.-s. Wang, and X.-f. Xu, “Photon-subtracted squeezed thermal state: nonclassicality and decoherence,” Phys. Rev. A 82, 043842 (2010).
[CrossRef]

C. Invernizzi, S. Olivares, M. G. A. Paris, and K. Banaszek, “Effect of noise and enhancement of nonlocality in on/off photodetection,” Phys. Rev. A 72, 042105 (2005).
[CrossRef]

M. Sasaki and S. Suzuki, “Multimode theory of measurement-induced non-Gaussian operation on wideband squeezed light: analytical formula,” Phys. Rev. A 73, 043807 (2006).
[CrossRef]

F. Dell’Anno, S. De Siena, L. Albano, and F. Illuminati, “Continuous-variable quantum teleportation with non-Gaussian resources,” Phys. Rev. A 76, 022301 (2007).
[CrossRef]

W. Schleich, D. F. Walls, and J. A. Wheeler, “Area of overlap and interference in phase space versus Wigner pseudoprobabilities,” Phys. Rev. A 38, 1177–1186 (1988).
[CrossRef]

Phys. Rev. D (1)

G. S. Agarwal and E. Wolf, “Calculus for functions of noncommuting operators and general phase-space methods in quantum mechanics. I. mapping theorems and ordering of functions of noncommuting operators,” Phys. Rev. D 2, 2161–2186(1970).
[CrossRef]

Phys. Rev. Lett. (5)

J. Wenger, R. Tualle-Brouri, and P. Grangier, “Non-Gaussian statistics from individual pulses of squeezed light,” Phys. Rev. Lett. 92, 153601 (2004).
[CrossRef]

J. Eisert, S. Scheel, and M. B. Plenio, “Distilling Gaussian states with Gaussian operations is impossible,” Phys. Rev. Lett. 89, 137903 (2002).
[CrossRef]

S. L. Braunstein and H. J. Kimble, “Teleportation of continuous quantum variables,” Phys. Rev. Lett. 80, 869–872 (1998).
[CrossRef]

N. J. Cerf, O. Krüger, P. Navez, R. F. Werner, and M. M. Wolf, “Non-Gaussian cloning of quantum coherent states is optimal,” Phys. Rev. Lett. 95, 070501 (2005).
[CrossRef]

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

Prog. Inform. (1)

J. S. Neergaard-Nielsen, M. Takeuchi, K. Wakui, H. Takahashi, K. Hayasaka, M. Takeoka, and M. Sasaki, “Photon subtraction from traveling fields-recent experimental demonstrations,” Prog. Inform. 8, 5–18 (2011).
[CrossRef]

Other (7)

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

P. Schleich Wolfgang, Quantum Optics in Phase Space(Wiley-VCH, 2001).

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

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

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

W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer, 1996).

R. R. Puri, Mathematical Methods of Quantum Optics (Springer-Verlag, 2001).

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Figures (5)

Fig. 1.
Fig. 1.

Squeezing degree Sm(r) of the m-PSSVS as the function of r for different values of m.

Fig. 2.
Fig. 2.

Photon-number distributions of the m-PSSVS for (a) m=2 and r=0.3, (b) m=2 and r=0.8, (c) m=5 and r=0.3, and (d) m=5 and r=0.8.

Fig. 3.
Fig. 3.

WFs for the m-PSSVS in phase space: (a) m=1 and r=0.3, (b) m=2 and r=0.3, (c) m=5 and r=0.3, (d) m=8 and r=0.3, (e) m=2 and r=0.8, and (f) m=5 and r=0.8.

Fig. 4.
Fig. 4.

WF evolution of the m-PSSVS with given n¯=0.05 for different values of m, r,and κtt in the thermal channel. (a) m=2, r=0.3, and κtt=0.1, (b) m=2, r=0.3, and κtt=0.5, (c) m=5, r=0.3, and κtt=0.1, (d) m=5, r=0.3, and κtt=0.5, (e) m=2, r=0.8, and κtt=0.1, and (f) m=5, r=0.8, and κtt=0.1.

Fig. 5.
Fig. 5.

WF evolution of the m-PSSVS for different values of m, r,and κpt in phase-damping channel. (a) m=2, r=0.3 and κpt=0.1, (b) m=2, r=0.3, and κpt=3, (c) m=3, r=0.3, and κpt=0.1, (d) m=3, r=0.3, and κpt=3, (e) m=17, r=0.3, and κpt=0.1, (f) m=17, r=0.3, and κpt=3, (g) m=2, r=0.8, and κpt=0.1, and (h) m=2, r=0.8, and κpt=3.

Equations (45)

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

S(r)|0=sech1/2rexp(12a2tanhr)|0,
amS(r)|0|r,m.
Nm=sechr0|exp(12a2tanhr)Hm,m(a,a)exp(12a2tanhr)|0.
Nm=2mτmsmexp[14(τ2+s2)sinh2r+τssinh2r]|τ=s=0.
xmn=0[m/2]m!22n(n!)2(m2n)!(11x2)n=Pm(x),
Nm=m!(isinhr)mPm(isinhr).
Sm=2|a2a2|+2aa2|a|2,
Sm(r)=2Nm|l=0[m/2]m!(m+2)!sinh2(m+1)rl!(l+1)!(m2l)!(12tanhr)2l+1|+2Nm+1Nm.
Pm(n)=Nm1n|amS(r)|00|S(r)am|n.
ρs=sechrexp[12(a2+a2)tanhraa],
ρs=1isinhrexp[12tanhr(a2+a2)+aa],
Pm(n)=1iNmsinhrn|amexp[12tanhr(a2+a2)+aa]am|n.
Pm(n)={[(2l)!]2tanh2lrsechr22l(l!)2(2lm)!m!(isinhr)mPm(isinhr)cm+n=2l0m+n=2l+1l=0,1,2,
Pm=0(n)={(2l)!22l(l!)2tanh2lrsechrn=2l0n=2l+1,
Pm=1(n)={(2l)!22l1l!(l1)!tanh2(l1)rsech3rn=2l10n=2l,
Wm(α,α*)=Nm10|S1(r)amΔ(α,α*)amS(r)|0,
amS(r)|0=S(r)bm|0,
Δ(erp,erx)=d2zπ2|β+zβz|eβz*zβ*,
Wm(α,α*)=Nm10|bmd2zπ2|β+zβz|eβz*zβ*bm|0.
|xη,ξ=1[π(η2+ξ2)]1/4exp[x22(η2+ξ2)+2axηiξη+iξ2(ηiξ)a2]|0.
(ηX+ξP)|xη,ξ=x|xη,ξ,
(ρa+νa)n=(iρν2)nHn(iρ2νa+iν2ρa).
bm|0=(isinh2r4)mHm(itanhr2a)|0.
Wm(α,α*)=1πNm(sinh2r4)mexp(2|β|2)n=0m2n(m!)2(tanhr)nn![(mn)!]2|Hmn(i2tanhrβ)|2,
W1(α,α*)=1π(4|β|21)exp(2|β|2).
dρ(t)dt=κt(n¯+1)[2aρ(t)aaaρ(t)ρ(t)aa]+κtn¯[2aρ(t)aaaρ(t)ρ(t)aa],
ρ(t)=(1κtn¯T1)i,j=0[κt(n¯+1)]i(κtn¯)ji!j!T1i+jajeaalnT2aiρ0aieaalnT2aj,
ρpt(t)=i(1κtn¯T1)Nmsinhri,j=0[κt(n¯+1)]i(κtn¯)ji!j!T1i+jajeaalnT2am+iexp[12tanhr(a2+a2)+aa]am+ieaalnT2aj.
ρpt(t)=iB(1κtn¯T1)Nmsinhrl=0m(m!)2[κt(n¯+1)T1]lBml![(ml)!]2(cothr2)mlHml[Btanhr2T2(κt(n¯+1)T1a+acothr)]Hml[Btanhr2T2(κt(n¯+1)T1a+acothr)]exp{[κtn¯T1Bκt(n¯+1)T1T221]aaBT222tanhr(a2+a2)},
ρpt(t)=im!B(1κtn¯T1)[Bκt(n¯+1)T1]mNmsinhrLm[BT22(κt(n¯+1)T1a+acothr)(κt(n¯+1)T1a+acothr)κt(n¯+1)T1]exp{[κtn¯T1Bκt(n¯+1)T1T221]aaBT222tanhr(a2+a2)},
ρpt(t;κtn¯0)=im!Cm+1/2(e2κtt1)mNmsinhrLm[Ce2κtt[(1e2κtt)a+acothr][(1e2κtt)a+acothr](1e2κtt)]exp{[(e2κtt1)e2κtt1]aaCe2κtt2tanhr(a2+a2)},
ρpt(t;m=0)=iB(1κtn¯T1)sinhrexp{[κtn¯T1Bκt(n¯+1)T1T221]aaBT222tanhr(a2+a2)},
Wpt(α,t)=Tr[ρpt(t)Δ(α,α*)].
Wpt(α,t)=Wm(α,t)Ws(α,t),
Ws(α,t)=B(1κtn¯T1)πh1sinhrexp[(4Dh1+2)|α|2+4Eh1(α2+α*2)],
Wm(α,t)=1Nml=0mn=0ml(1)mn(m!)2[κt(n¯+1)T1]lBml!n![(mln)!]2(h2cothr2)ml(h3h2)n|Hmln(h42h2)|2,
D=Bκt(n¯+1)T1T22κtn¯T11,E=BT222tanhr,h1=D24E2,h2=2BT22h1[Dκt(n¯+1)T1+Eκt2(n¯+1)2T12tanhr+Ecothr]1,h3=2BT22h1[Dκt2(n¯+1)2T12tanhr+Dcothr+4Eκt(n¯+1)T1],h4=22BtanhrT2h1[(Dκt(n¯+1)T1+2Ecothr)α+(Dcothr+2Eκt(n¯+1)T1)α*].
Wpt(α,κtt)=1π(2n¯+1)exp(22n¯+1|α|2),
dρ(t)dt=κp[2Aρ(t)AAAρ(t)ρ(t)AA],
ρ(t)=eκptA2n=0(2κpt)nn!Anρ0AneκptA2,
ρpp(t)=sechrNmk,k=0(2k)!(2k)!k!k!(2km)!(2km)!(tanhr2)k+ke4κpt(kk)2a2kma2kmexp(aa),
ρpp(t;m=0)=sechrk,k=01k!k!(tanhr2)k+ke4κpt(kk)2a2ka2kexp(aa),
ρpp(κpt)=sechrNmk=0[(2k)!]2(k!)2[(2km)!]2(tanhr2)2k(aa)2kmexp(aa),
Wpp(α,t)=sechrπNmk,k=0(2k)!(2k)!k!k!(2km)!(2km)!(tanhr2)k+ke4κpt(kk)2exp(2|α|2)H2km,2km(2α,2α*),
Wpp(α,κpt)=sechrNmk=0[(2k)!]2(k!)2(2km)!(tanhr2)2kW|2km2km|(α),

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