Abstract

We investigate statistical properties of photon-subtracted squeezed vacuum (PSSV) in a thermal environment with average thermal photon number n¯ and dissipative coefficient κ by deriving the analytical expressions of the density operator and the Wigner function (WF). The normalization of m-PSSV is proved to be the m-order Lagendre polynomials. Time evolution of the photocount distribution is also related to Lagendre polynomials. The tomogram of PSSV is calculated by using the intermediate coordinate-momentum representation. Especially, the nonclassicality is discussed by virtue of the negativity of WF. It is found that the WF is always negative for all squeezing parameters λ if κt<12ln(2n¯+2)(2n¯+1) for the single-PSSV.

© 2008 Optical Society of America

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  5. H. Jeong, A. P. Lund, and T. C. Ralph, “Production of superpositions of coherent states in traveling optical fields with inefficient photon detection,” Phys. Rev. A 72, 013801 (2005).
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  6. J. S. Neergaard-Nielsen, B. Melholt Nielsen, C. Hettich, K. Mølmer, and E. S. Polzik, “Generation of a superposition of odd photon number states for quantum information networks,” Phys. Rev. Lett. 97, 083604 (2006).
    [CrossRef] [PubMed]
  7. A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, and Ph. Grangier, “Increasing entanglement between Gaussian states by coherent photon subtraction,” Phys. Rev. Lett. 98, 030502 (2007).
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  8. A. Ourjoumtsev, R. Tualle-Brouri, and Ph. Grangier, “Quantum homodyne tomography of a two-photon Fock state,” Phys. Rev. Lett. 96, 213601 (2006).
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    [CrossRef] [PubMed]
  12. A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and Ph. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science 312, 83-86 (2006).
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  26. 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]
  27. 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]
  28. C. Gardiner and P. Zoller, Quantum Noise (Springer-Verlag, 2000).
  29. R. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529-2539 (1963).
    [CrossRef]
  30. R. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131, 2766-2788 (1963).
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    [CrossRef]
  34. H.-Y. Fan, H.-L. Lu, and Y. Fan, “Newton-Leibniz integration for ket-bra operators in quantum mechanics and derivation of entangled state representations,” Ann. Phys. 321, 480-494 (2006).
    [CrossRef]
  35. A. Wünsche, “On the IWOP technique in quantum optics,” J. Opt. B: Quantum Semiclassical Opt. 1, R11-R21 (1999).
    [CrossRef]
  36. H. Y. Fan and L. Y. Hu, “Two quantum-mechanical photocount formulas,” Opt. Lett. 33, 443-445 (2008).
    [CrossRef] [PubMed]
  37. H. Y. Fan and H. L. Chen, “Two-parameter Radon transformation of the Wigner operator and its inverse,” Chin. Phys. Lett. 18, 850-853 (2001).
  38. H. Y. Fan and L. Y. Hu, “New approach for analyzing time evolution of density operator in a dissipative channel by the entangled state representation,” Opt. Commun. 281, 5571-5573 (2008).
    [CrossRef]
  39. 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]
  40. I. S. Gradshteyn and L. M. Ryzhik, Tables of Integration Series and Products (Academic, 1980).
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2008 (6)

2007 (3)

K. Wakui, H. Takahashi, A. Furusawa, and M. Sasaki, “Controllable generation of highly nonclassical states from nearly pure squeezed vacua,” Opt. Express 15, 3568-3574 (2007).
[CrossRef] [PubMed]

A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, and Ph. Grangier, “Increasing entanglement between Gaussian states by coherent photon subtraction,” Phys. Rev. Lett. 98, 030502 (2007).
[CrossRef] [PubMed]

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

2006 (5)

J. S. Neergaard-Nielsen, B. Melholt Nielsen, C. Hettich, K. Mølmer, and E. S. Polzik, “Generation of a superposition of odd photon number states for quantum information networks,” Phys. Rev. Lett. 97, 083604 (2006).
[CrossRef] [PubMed]

A. Ourjoumtsev, R. Tualle-Brouri, and Ph. Grangier, “Quantum homodyne tomography of a two-photon Fock state,” Phys. Rev. Lett. 96, 213601 (2006).
[CrossRef] [PubMed]

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and Ph. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science 312, 83-86 (2006).
[CrossRef] [PubMed]

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, H.-L. Lu, and Y. Fan, “Newton-Leibniz integration for ket-bra operators in quantum mechanics and derivation of entangled state representations,” Ann. Phys. 321, 480-494 (2006).
[CrossRef]

2005 (4)

S. Olivares and M. G. A. Paris, “Squeezed Fock state by inconclusive photon subtraction,” J. Opt. B: Quantum Semiclassical Opt. 7, S616-S621 (2005).
[CrossRef]

S. Olivares and M. G. A. Paris, “Photon subtracted states and enhancement of nonlocality in the presence of noise,” J. Opt. B: Quantum Semiclassical Opt. 7, S392-S397 (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]

H. Jeong, A. P. Lund, and T. C. Ralph, “Production of superpositions of coherent states in traveling optical fields with inefficient photon detection,” Phys. Rev. A 72, 013801 (2005).
[CrossRef]

2004 (2)

H. Nha and H. J. Carmichael, “Proposed test of quantum nonlocality for continuous variables,” Phys. Rev. Lett. 93, 020401 (2004).
[CrossRef] [PubMed]

R. García-Patrón, J. Fiurášek, N. J. Cerf, J. Wenger, R. Tualle-Brouri, and P. Grangier, “Proposal for a loophole-free bell test using homodyne detection,” Phys. Rev. Lett. 93, 130409 (2004).
[CrossRef] [PubMed]

2003 (1)

D. E. Browne, J. Eisert, S. Scheel, and M. B. Plenio, “Driving non-Gaussian to Gaussian states with linear optics,” Phys. Rev. A 67, 062320 (2003).
[CrossRef]

2002 (1)

S. D. Bartlett and B. C. Sanders, “Universal continuous-variable quantum computation: requirement of optical nonlinearity for photon counting,” Phys. Rev. A 65, 042304 (2002).
[CrossRef]

2001 (1)

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

1999 (1)

A. Wünsche, “On the IWOP technique in quantum optics,” J. Opt. B: Quantum Semiclassical Opt. 1, R11-R21 (1999).
[CrossRef]

1998 (1)

F. De Martini, “Amplification of quantum entanglement,” Phys. Rev. Lett. 81, 2842-2845 (1998).
[CrossRef]

1997 (1)

M. Dakna, T. Anhut, T. Opatrny, L. Knoll, and D.-G. Welsch, “Generating Schrödinger-cat-like states by means of conditional measurements on a beam splitter,” Phys. Rev. A 55, 3184-3194 (1997).
[CrossRef]

1993 (1)

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244-1247 (1993).
[CrossRef] [PubMed]

1992 (1)

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] [PubMed]

1989 (2)

K. Vogel and H. Risken, “Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase,” Phys. Rev. A 40, 2847-2849 (1989).
[CrossRef] [PubMed]

H. Y. Fan and V. J. Linde, “Similarity transformations in one- and two-mode Fock space,” J. Phys. A 24, 2529-2538 (1989).
[CrossRef]

1964 (1)

P. L. Kelley and W. H. Kleiner, “Theory of electromagnetic field measurement and photoelectron counting,” Phys. Rev. 136, 316-334 (1964).
[CrossRef]

1963 (2)

R. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529-2539 (1963).
[CrossRef]

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]

Anhut, T.

M. Dakna, T. Anhut, T. Opatrny, L. Knoll, and D.-G. Welsch, “Generating Schrödinger-cat-like states by means of conditional measurements on a beam splitter,” Phys. Rev. A 55, 3184-3194 (1997).
[CrossRef]

Bartlett, S. D.

S. D. Bartlett and B. C. Sanders, “Universal continuous-variable quantum computation: requirement of optical nonlinearity for photon counting,” Phys. Rev. A 65, 042304 (2002).
[CrossRef]

Beck, M.

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244-1247 (1993).
[CrossRef] [PubMed]

Biswas, A.

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

Bouwmeester, D.

D. Bouwmeester, A. Ekert, and A. Zeilinger, The Physics of Quantum Information (Springer-Verlag, 2000).

Browne, D. E.

D. E. Browne, J. Eisert, S. Scheel, and M. B. Plenio, “Driving non-Gaussian to Gaussian states with linear optics,” Phys. Rev. A 67, 062320 (2003).
[CrossRef]

Bužek, V.

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] [PubMed]

Bychkov, Yu. A.

A. P. Prudnikov, Yu. A. Bychkov, and O. I. Marychev, Integrals and Series: Volume 2: Special Functions (Nauka, 1983), Subsection 5.12.2.

Carmichael, H. J.

H. Nha and H. J. Carmichael, “Proposed test of quantum nonlocality for continuous variables,” Phys. Rev. Lett. 93, 020401 (2004).
[CrossRef] [PubMed]

Cerf, N. J.

R. García-Patrón, J. Fiurášek, N. J. Cerf, J. Wenger, R. Tualle-Brouri, and P. Grangier, “Proposal for a loophole-free bell test using homodyne detection,” Phys. Rev. Lett. 93, 130409 (2004).
[CrossRef] [PubMed]

Chen, H. L.

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

Dakna, M.

M. Dakna, T. Anhut, T. Opatrny, L. Knoll, and D.-G. Welsch, “Generating Schrödinger-cat-like states by means of conditional measurements on a beam splitter,” Phys. Rev. A 55, 3184-3194 (1997).
[CrossRef]

Dantan, A.

A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, and Ph. Grangier, “Increasing entanglement between Gaussian states by coherent photon subtraction,” Phys. Rev. Lett. 98, 030502 (2007).
[CrossRef] [PubMed]

De Martini, F.

F. De Martini, “Amplification of quantum entanglement,” Phys. Rev. Lett. 81, 2842-2845 (1998).
[CrossRef]

de Vasconcelos, H. M.

Eisert, J.

D. E. Browne, J. Eisert, S. Scheel, and M. B. Plenio, “Driving non-Gaussian to Gaussian states with linear optics,” Phys. Rev. A 67, 062320 (2003).
[CrossRef]

Ekert, A.

D. Bouwmeester, A. Ekert, and A. Zeilinger, The Physics of Quantum Information (Springer-Verlag, 2000).

Fan, H. Y.

H. Y. Fan and L. Y. Hu, “Two quantum-mechanical photocount formulas,” Opt. Lett. 33, 443-445 (2008).
[CrossRef] [PubMed]

H. Y. Fan and L. Y. Hu, “New approach for analyzing time evolution of density operator in a dissipative channel by the entangled state representation,” Opt. Commun. 281, 5571-5573 (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]

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. Chen, “Two-parameter Radon transformation of the Wigner operator and its inverse,” Chin. Phys. Lett. 18, 850-853 (2001).

H. Y. Fan and V. J. Linde, “Similarity transformations in one- and two-mode Fock space,” J. Phys. A 24, 2529-2538 (1989).
[CrossRef]

Fan, H.-Y.

H.-Y. Fan, H.-L. Lu, and Y. Fan, “Newton-Leibniz integration for ket-bra operators in quantum mechanics and derivation of entangled state representations,” Ann. Phys. 321, 480-494 (2006).
[CrossRef]

Fan, Y.

H.-Y. Fan, H.-L. Lu, and Y. Fan, “Newton-Leibniz integration for ket-bra operators in quantum mechanics and derivation of entangled state representations,” Ann. Phys. 321, 480-494 (2006).
[CrossRef]

Faridani, A.

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244-1247 (1993).
[CrossRef] [PubMed]

Fiurášek, J.

R. García-Patrón, J. Fiurášek, N. J. Cerf, J. Wenger, R. Tualle-Brouri, and P. Grangier, “Proposal for a loophole-free bell test using homodyne detection,” Phys. Rev. Lett. 93, 130409 (2004).
[CrossRef] [PubMed]

Furusawa, A.

García-Patrón, R.

R. García-Patrón, J. Fiurášek, N. J. Cerf, J. Wenger, R. Tualle-Brouri, and P. Grangier, “Proposal for a loophole-free bell test using homodyne detection,” Phys. Rev. Lett. 93, 130409 (2004).
[CrossRef] [PubMed]

Gardiner, C.

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

Glancy, S.

Glauber, R.

R. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529-2539 (1963).
[CrossRef]

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

Gradshteyn, I. S.

I. S. Gradshteyn and L. M. Ryzhik, Tables of Integration Series and Products (Academic, 1980).

Grangier, P.

R. García-Patrón, J. Fiurášek, N. J. Cerf, J. Wenger, R. Tualle-Brouri, and P. Grangier, “Proposal for a loophole-free bell test using homodyne detection,” Phys. Rev. Lett. 93, 130409 (2004).
[CrossRef] [PubMed]

Grangier, Ph.

A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, and Ph. Grangier, “Increasing entanglement between Gaussian states by coherent photon subtraction,” Phys. Rev. Lett. 98, 030502 (2007).
[CrossRef] [PubMed]

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and Ph. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science 312, 83-86 (2006).
[CrossRef] [PubMed]

A. Ourjoumtsev, R. Tualle-Brouri, and Ph. Grangier, “Quantum homodyne tomography of a two-photon Fock state,” Phys. Rev. Lett. 96, 213601 (2006).
[CrossRef] [PubMed]

Guo, G.-C.

Hettich, C.

J. S. Neergaard-Nielsen, B. Melholt Nielsen, C. Hettich, K. Mølmer, and E. S. Polzik, “Generation of a superposition of odd photon number states for quantum information networks,” Phys. Rev. Lett. 97, 083604 (2006).
[CrossRef] [PubMed]

Hu, L. Y.

H. Y. Fan and L. Y. Hu, “Two quantum-mechanical photocount formulas,” Opt. Lett. 33, 443-445 (2008).
[CrossRef] [PubMed]

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]

H. Y. Fan and L. Y. Hu, “New approach for analyzing time evolution of density operator in a dissipative channel by the entangled state representation,” Opt. Commun. 281, 5571-5573 (2008).
[CrossRef]

Jeong, H.

H. Jeong, J. Lee, and H. Nha, “Decoherence of highly mixed macroscopic quantum superpositions,” J. Opt. Soc. Am. B 25, 1025-1030 (2008).
[CrossRef]

H. Jeong, A. P. Lund, and T. C. Ralph, “Production of superpositions of coherent states in traveling optical fields with inefficient photon detection,” Phys. Rev. A 72, 013801 (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]

Kelley, P. L.

P. L. Kelley and W. H. Kleiner, “Theory of electromagnetic field measurement and photoelectron counting,” Phys. Rev. 136, 316-334 (1964).
[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]

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] [PubMed]

Klauder, J. R.

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

Kleiner, W. H.

P. L. Kelley and W. H. Kleiner, “Theory of electromagnetic field measurement and photoelectron counting,” Phys. Rev. 136, 316-334 (1964).
[CrossRef]

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]

Knoll, L.

M. Dakna, T. Anhut, T. Opatrny, L. Knoll, and D.-G. Welsch, “Generating Schrödinger-cat-like states by means of conditional measurements on a beam splitter,” Phys. Rev. A 55, 3184-3194 (1997).
[CrossRef]

Laurat, J.

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and Ph. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science 312, 83-86 (2006).
[CrossRef] [PubMed]

Lee, J.

Li, S.-B.

Linde, V. J.

H. Y. Fan and V. J. Linde, “Similarity transformations in one- and two-mode Fock space,” J. Phys. A 24, 2529-2538 (1989).
[CrossRef]

Liu, J.

Lu, H.-L.

H.-Y. Fan, H.-L. Lu, and Y. Fan, “Newton-Leibniz integration for ket-bra operators in quantum mechanics and derivation of entangled state representations,” Ann. Phys. 321, 480-494 (2006).
[CrossRef]

Lund, A. P.

H. Jeong, A. P. Lund, and T. C. Ralph, “Production of superpositions of coherent states in traveling optical fields with inefficient photon detection,” Phys. Rev. A 72, 013801 (2005).
[CrossRef]

Marychev, O. I.

A. P. Prudnikov, Yu. A. Bychkov, and O. I. Marychev, Integrals and Series: Volume 2: Special Functions (Nauka, 1983), Subsection 5.12.2.

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]

Milburn, G. J.

D. F. Walls and G. J. Milburn, Quantum Optics (Springer-Verlag, 1994).

Mølmer, K.

J. S. Neergaard-Nielsen, B. Melholt Nielsen, C. Hettich, K. Mølmer, and E. S. Polzik, “Generation of a superposition of odd photon number states for quantum information networks,” Phys. Rev. Lett. 97, 083604 (2006).
[CrossRef] [PubMed]

Neergaard-Nielsen, J. S.

J. S. Neergaard-Nielsen, B. Melholt Nielsen, C. Hettich, K. Mølmer, and E. S. Polzik, “Generation of a superposition of odd photon number states for quantum information networks,” Phys. Rev. Lett. 97, 083604 (2006).
[CrossRef] [PubMed]

Nha, H.

H. Jeong, J. Lee, and H. Nha, “Decoherence of highly mixed macroscopic quantum superpositions,” J. Opt. Soc. Am. B 25, 1025-1030 (2008).
[CrossRef]

H. Nha and H. J. Carmichael, “Proposed test of quantum nonlocality for continuous variables,” Phys. Rev. Lett. 93, 020401 (2004).
[CrossRef] [PubMed]

Nielsen, B. Melholt

J. S. Neergaard-Nielsen, B. Melholt Nielsen, C. Hettich, K. Mølmer, and E. S. Polzik, “Generation of a superposition of odd photon number states for quantum information networks,” Phys. Rev. Lett. 97, 083604 (2006).
[CrossRef] [PubMed]

Olivares, S.

S. Olivares and M. G. A. Paris, “Squeezed Fock state by inconclusive photon subtraction,” J. Opt. B: Quantum Semiclassical Opt. 7, S616-S621 (2005).
[CrossRef]

S. Olivares and M. G. A. Paris, “Photon subtracted states and enhancement of nonlocality in the presence of noise,” J. Opt. B: Quantum Semiclassical Opt. 7, S392-S397 (2005).
[CrossRef]

Opatrny, T.

M. Dakna, T. Anhut, T. Opatrny, L. Knoll, and D.-G. Welsch, “Generating Schrödinger-cat-like states by means of conditional measurements on a beam splitter,” Phys. Rev. A 55, 3184-3194 (1997).
[CrossRef]

Ourjoumtsev, A.

A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, and Ph. Grangier, “Increasing entanglement between Gaussian states by coherent photon subtraction,” Phys. Rev. Lett. 98, 030502 (2007).
[CrossRef] [PubMed]

A. Ourjoumtsev, R. Tualle-Brouri, and Ph. Grangier, “Quantum homodyne tomography of a two-photon Fock state,” Phys. Rev. Lett. 96, 213601 (2006).
[CrossRef] [PubMed]

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and Ph. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science 312, 83-86 (2006).
[CrossRef] [PubMed]

Paris, M. G. A.

S. Olivares and M. G. A. Paris, “Photon subtracted states and enhancement of nonlocality in the presence of noise,” J. Opt. B: Quantum Semiclassical Opt. 7, S392-S397 (2005).
[CrossRef]

S. Olivares and M. G. A. Paris, “Squeezed Fock state by inconclusive photon subtraction,” J. Opt. B: Quantum Semiclassical Opt. 7, S616-S621 (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]

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D. E. Browne, J. Eisert, S. Scheel, and M. B. Plenio, “Driving non-Gaussian to Gaussian states with linear optics,” Phys. Rev. A 67, 062320 (2003).
[CrossRef]

Polzik, E. S.

J. S. Neergaard-Nielsen, B. Melholt Nielsen, C. Hettich, K. Mølmer, and E. S. Polzik, “Generation of a superposition of odd photon number states for quantum information networks,” Phys. Rev. Lett. 97, 083604 (2006).
[CrossRef] [PubMed]

Prudnikov, A. P.

A. P. Prudnikov, Yu. A. Bychkov, and O. I. Marychev, Integrals and Series: Volume 2: Special Functions (Nauka, 1983), Subsection 5.12.2.

Puri, R. R.

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

Ralph, T. C.

H. Jeong, A. P. Lund, and T. C. Ralph, “Production of superpositions of coherent states in traveling optical fields with inefficient photon detection,” Phys. Rev. A 72, 013801 (2005).
[CrossRef]

Raymer, M. G.

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244-1247 (1993).
[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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I. S. Gradshteyn and L. M. Ryzhik, Tables of Integration Series and Products (Academic, 1980).

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S. D. Bartlett and B. C. Sanders, “Universal continuous-variable quantum computation: requirement of optical nonlinearity for photon counting,” Phys. Rev. A 65, 042304 (2002).
[CrossRef]

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Scheel, S.

D. E. Browne, J. Eisert, S. Scheel, and M. B. Plenio, “Driving non-Gaussian to Gaussian states with linear optics,” Phys. Rev. A 67, 062320 (2003).
[CrossRef]

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M. O. Scully and M. S. Zubairy, Quantum Optics (Cambidge U. Press, 1997).

Skargerstam, B. S.

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

Smithey, D. T.

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244-1247 (1993).
[CrossRef] [PubMed]

Takahashi, H.

Tualle-Brouri, R.

A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, and Ph. Grangier, “Increasing entanglement between Gaussian states by coherent photon subtraction,” Phys. Rev. Lett. 98, 030502 (2007).
[CrossRef] [PubMed]

A. Ourjoumtsev, R. Tualle-Brouri, and Ph. Grangier, “Quantum homodyne tomography of a two-photon Fock state,” Phys. Rev. Lett. 96, 213601 (2006).
[CrossRef] [PubMed]

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and Ph. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science 312, 83-86 (2006).
[CrossRef] [PubMed]

R. García-Patrón, J. Fiurášek, N. J. Cerf, J. Wenger, R. Tualle-Brouri, and P. Grangier, “Proposal for a loophole-free bell test using homodyne detection,” Phys. Rev. Lett. 93, 130409 (2004).
[CrossRef] [PubMed]

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K. Vogel and H. Risken, “Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase,” Phys. Rev. A 40, 2847-2849 (1989).
[CrossRef] [PubMed]

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D. F. Walls and G. J. Milburn, Quantum Optics (Springer-Verlag, 1994).

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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]

Welsch, D.-G.

M. Dakna, T. Anhut, T. Opatrny, L. Knoll, and D.-G. Welsch, “Generating Schrödinger-cat-like states by means of conditional measurements on a beam splitter,” Phys. Rev. A 55, 3184-3194 (1997).
[CrossRef]

Wenger, J.

R. García-Patrón, J. Fiurášek, N. J. Cerf, J. Wenger, R. Tualle-Brouri, and P. Grangier, “Proposal for a loophole-free bell test using homodyne detection,” Phys. Rev. Lett. 93, 130409 (2004).
[CrossRef] [PubMed]

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A. Wünsche, “On the IWOP technique in quantum optics,” J. Opt. B: Quantum Semiclassical Opt. 1, R11-R21 (1999).
[CrossRef]

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D. Bouwmeester, A. Ekert, and A. Zeilinger, The Physics of Quantum Information (Springer-Verlag, 2000).

Zoller, P.

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

Zou, X.-B.

Zubairy, M. S.

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambidge U. Press, 1997).

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H.-Y. Fan, H.-L. Lu, and Y. Fan, “Newton-Leibniz integration for ket-bra operators in quantum mechanics and derivation of entangled state representations,” Ann. Phys. 321, 480-494 (2006).
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H. Y. Fan and H. L. Chen, “Two-parameter Radon transformation of the Wigner operator and its inverse,” Chin. Phys. Lett. 18, 850-853 (2001).

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. Opt. B: Quantum Semiclassical Opt. (3)

A. Wünsche, “On the IWOP technique in quantum optics,” J. Opt. B: Quantum Semiclassical Opt. 1, R11-R21 (1999).
[CrossRef]

S. Olivares and M. G. A. Paris, “Squeezed Fock state by inconclusive photon subtraction,” J. Opt. B: Quantum Semiclassical Opt. 7, S616-S621 (2005).
[CrossRef]

S. Olivares and M. G. A. Paris, “Photon subtracted states and enhancement of nonlocality in the presence of noise,” J. Opt. B: Quantum Semiclassical Opt. 7, S392-S397 (2005).
[CrossRef]

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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).
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H. Y. Fan and L. Y. Hu, “New approach for analyzing time evolution of density operator in a dissipative channel by the entangled state representation,” Opt. Commun. 281, 5571-5573 (2008).
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Opt. Express (1)

Opt. Lett. (1)

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[CrossRef]

Phys. Rev. A (8)

S. D. Bartlett and B. C. Sanders, “Universal continuous-variable quantum computation: requirement of optical nonlinearity for photon counting,” Phys. Rev. A 65, 042304 (2002).
[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]

M. Dakna, T. Anhut, T. Opatrny, L. Knoll, and D.-G. Welsch, “Generating Schrödinger-cat-like states by means of conditional measurements on a beam splitter,” Phys. Rev. A 55, 3184-3194 (1997).
[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] [PubMed]

D. E. Browne, J. Eisert, S. Scheel, and M. B. Plenio, “Driving non-Gaussian to Gaussian states with linear optics,” Phys. Rev. A 67, 062320 (2003).
[CrossRef]

K. Vogel and H. Risken, “Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase,” Phys. Rev. A 40, 2847-2849 (1989).
[CrossRef] [PubMed]

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

H. Jeong, A. P. Lund, and T. C. Ralph, “Production of superpositions of coherent states in traveling optical fields with inefficient photon detection,” Phys. Rev. A 72, 013801 (2005).
[CrossRef]

Phys. Rev. Lett. (7)

J. S. Neergaard-Nielsen, B. Melholt Nielsen, C. Hettich, K. Mølmer, and E. S. Polzik, “Generation of a superposition of odd photon number states for quantum information networks,” Phys. Rev. Lett. 97, 083604 (2006).
[CrossRef] [PubMed]

A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, and Ph. Grangier, “Increasing entanglement between Gaussian states by coherent photon subtraction,” Phys. Rev. Lett. 98, 030502 (2007).
[CrossRef] [PubMed]

A. Ourjoumtsev, R. Tualle-Brouri, and Ph. Grangier, “Quantum homodyne tomography of a two-photon Fock state,” Phys. Rev. Lett. 96, 213601 (2006).
[CrossRef] [PubMed]

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244-1247 (1993).
[CrossRef] [PubMed]

H. Nha and H. J. Carmichael, “Proposed test of quantum nonlocality for continuous variables,” Phys. Rev. Lett. 93, 020401 (2004).
[CrossRef] [PubMed]

R. García-Patrón, J. Fiurášek, N. J. Cerf, J. Wenger, R. Tualle-Brouri, and P. Grangier, “Proposal for a loophole-free bell test using homodyne detection,” Phys. Rev. Lett. 93, 130409 (2004).
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[CrossRef]

Science (1)

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and Ph. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science 312, 83-86 (2006).
[CrossRef] [PubMed]

Other (8)

D. Bouwmeester, A. Ekert, and A. Zeilinger, The Physics of Quantum Information (Springer-Verlag, 2000).

D. F. Walls and G. J. Milburn, Quantum Optics (Springer-Verlag, 1994).

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambidge U. Press, 1997).

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

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

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

I. S. Gradshteyn and L. M. Ryzhik, Tables of Integration Series and Products (Academic, 1980).

A. P. Prudnikov, Yu. A. Bychkov, and O. I. Marychev, Integrals and Series: Volume 2: Special Functions (Nauka, 1983), Subsection 5.12.2.

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Figures (6)

Fig. 1
Fig. 1

Q-parameter as a function of squeezing parameter λ for different m = 1 , 2, 3, 4, 29, and 30.

Fig. 2
Fig. 2

Variation of P in phase space for λ = 0.3 , n ¯ = 1 , and κ t = 0.05 .

Fig. 3
Fig. 3

WFs of single-photon-subtracted squeezed vacuum states in phase space for λ = 0.3 , n ¯ = 1 at (a) κ t = 0.05 , (b) κ t = 0.1 , (c) κ t = 0.2 , and (d) κ t = 0.5 .

Fig. 4
Fig. 4

WFs of SPSSV states in phase space for λ = 0.3 and κ t = 0.05 with (a) n ¯ = 0 , (b) n ¯ = 1 , (c) n ¯ = 2 , and (d) n ¯ = 10 .

Fig. 5
Fig. 5

WFs of SPSSV states in phase space for n ¯ = 1 and κ t = 0.05 with (a) λ = 0.03 , (b) λ = 0.5 , (c) λ = 0.8 , and (d) λ = 1.5 .

Fig. 6
Fig. 6

WFs of PSSV states in phase space for λ = 0.8 ; (a) κ t = 0 , m = 2 , (b) κ t = 0.2 , m = 2 , (c) κ t = 0 , m = 3 , and (d) κ t = 0.2 , m = 3 .

Equations (77)

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λ m = N λ , m a m S ( λ ) 0 ,
λ m = N λ , m S ( λ ) S ( λ ) a m S ( λ ) 0 = N λ , m S ( λ ) ( a cosh λ a sinh λ ) m 0 .
( a μ + ν a ) m = ( i μ ν 2 ) m : H m ( i μ 2 ν a + i ν 2 μ a ) : ,
( a cosh λ a sinh λ ) m = sinh m 2 2 λ 2 m : H m ( coth λ 2 a tanh λ 2 a ) : .
λ m = N λ , m S ( λ ) ( a cosh λ a sinh λ ) m 0 = sinh m 2 2 λ 2 m N λ , m S ( λ ) H m ( tanh λ 2 a ) 0 ,
H m ( x ) = m t m exp ( 2 x t t 2 ) t = 0 ,
1 = sinh m 2 λ 2 2 m N λ , m 2 0 H m ( tanh λ 2 a ) H m ( tanh λ 2 a ) 0 = sinh m 2 λ 2 2 m N λ , m 2 2 m t m τ m exp ( t 2 τ 2 ) 0 e 2 tanh λ a t e 2 tanh λ a τ 0 t , τ = 0 = sinh m 2 λ 2 2 m N λ , m 2 2 m t m τ m exp ( t 2 τ 2 + 2 τ t tanh λ ) t , τ = 0 ,
2 m t m τ m exp ( t 2 τ 2 + 2 x τ t ) t , τ = 0 = n , l , k = 0 ( ) n + l n ! l ! k ! ( 2 x ) k 2 m t m τ m τ 2 n + k t 2 l + k t , τ = 0 = 2 m m ! n = 0 [ m 2 ] m ! 2 2 n ( n ! ) 2 ( m 2 n ) ! x m 2 n ,
N λ , m 2 = m ! ( i sinh λ ) m ( i sinh λ ) m n = 0 [ m 2 ] m ! 2 2 n ( n ! ) 2 ( m 2 n ) ! ( 1 1 ( i sinh λ ) 2 ) n .
x m l = 0 [ m 2 ] m ! 2 2 l ( l ! ) 2 ( m 2 l ) ! ( 1 1 x 2 ) l = P m ( x ) ,
N λ , m 2 = m ! ( i sinh λ ) m P m ( i sinh λ ) .
λ m = [ m ! ( i sinh λ ) m P m ( i sinh λ ) ] 1 2 a m S ( λ ) 0 .
a a = N λ , m 2 0 S ( λ ) a m + 1 a m + 1 S ( λ ) 0 = N λ , m 2 N λ , m + 1 2 = ( m + 1 ) ( i sinh λ ) P m + 1 ( i sinh λ ) P m ( i sinh λ ) ,
a 2 a 2 = N λ , m 2 N λ , m + 2 2 = ( m + 2 ) ( m + 1 ) ( i sinh λ ) 2 P m + 2 ( i sinh λ ) P m ( i sinh λ ) ,
Q a 2 a 2 a a a a = ( i sinh λ ) { ( m + 2 ) P m + 2 ( i sinh λ ) P m + 1 ( i sinh λ ) ( m + 1 ) P m + 1 ( i sinh λ ) P m ( i sinh λ ) } .
d ρ d t = κ ( n ¯ + 1 ) ( 2 a ρ a a a ρ ρ a a ) + κ n ¯ ( 2 a ρ a a a ρ ρ a a ) ,
ρ λ , m ( t ) = e κ t + Γ 0 k , l = 0 Γ k Γ + l e 2 l Γ 0 k ! l ! G ( a , a , t , n ¯ ) ,
G ( a , a ; t , n ¯ ) = e Γ 0 a a a l a k ρ 0 a k a l a Γ 0 a a ,
T = 1 e 2 κ t , Γ + = n ¯ T n ¯ T + 1 , Γ = ( n ¯ + 1 ) T n ¯ T + 1 ,
Γ 0 = ln e κ t n ¯ T + 1 .
e Γ 0 a a a e Γ 0 a a = a e Γ 0 , e Γ 0 a a a e Γ 0 a a = a e Γ 0 ,
G ( a , a ; t , n ¯ ) = N λ , m 2 sech λ e Γ 0 a a a l a k + m e ( 1 2 ) a 2 tanh λ 0 0 e ( 1 2 ) a 2 tanh λ a k + m a l e Γ 0 a a = N λ , m 2 sech λ e 2 ( l k m ) Γ 0 a l a k + m e μ 2 a 2 0 0 e μ 2 a 2 a k + m a l ,
a k + m e μ 2 a 2 0 = e μ 2 a 2 ( a 2 μ 2 a ) k + m 0 = e μ 2 a 2 μ m + k H m + k ( μ a ) 0 .
G ( a , a ; t , n ¯ ) = N λ , m 2 2 m + k tanh m + k λ cosh λ : ( e 2 Γ 0 a a ) l e μ 2 ( a 2 + a 2 ) a a H m + k ( μ a ) H m + k ( μ a ) : .
ρ λ , m ( t ) = C λ , m ( t ) : e ( Γ + 1 ) a a e μ 2 ( a 2 + a 2 ) k = 0 Γ k tanh k λ 2 k k ! H m + k ( μ a ) H m + k ( μ a ) : ,
ρ λ , m ( t ) = 2 m C λ , m ( t ) : m γ m e ( Γ + 1 ) a a 1 γ 2 exp [ a 2 + a 2 2 γ a a 2 ( 1 γ 2 ) e 2 Γ 0 tanh λ ] : γ = Γ tanh λ .
ρ λ , m ( t ) = 2 m C λ , m ( t ) : m γ m e a a 1 γ 2 exp [ a 2 + a 2 2 γ a a 2 ( 1 γ 2 ) e 2 κ t tanh λ ] : γ = T tanh λ
ρ λ , 1 ( ) = 1 n ¯ + 1 : e [ 1 ( n ¯ + 1 ) ] a a 1 n ¯ + 1 e a a ln [ n ¯ ( n ¯ + 1 ) ] ,
W ( α , α * , t ) = 1 π 2 e 2 α 2 β ρ ( t ) β exp [ 2 ( β α * β * α ) ] d 2 β ,
W ( α , α * , t ) = 2 m e 2 α 2 π m γ m C λ , m ( t ) 1 γ 2 D 2 e 4 Γ 0 tanh 2 λ × exp [ 4 ( 1 γ 2 ) D α 2 + e 2 Γ 0 tanh λ Re α 2 D 2 e 4 Γ 0 tanh 2 λ ] γ = Γ tanh λ ,
d 2 z π exp ( ζ z 2 + ξ z + η z * + f z 2 + g z * 2 ) = 1 ζ 2 4 f g exp [ ζ ξ η + ξ 2 g + η 2 f ζ 2 4 f g ] .
W m = 1 ( α , α * , t ) = P π sech 3 λ Q 5 2 exp [ 2 Q ( R α 2 + 2 e 2 κ t tanh λ Re α 2 ) ] ,
R = ( 1 2 T ) tanh 2 λ + 2 T n ¯ sech 2 λ + 1 ,
Q = 4 T n ¯ + ( 4 T 2 n ¯ 2 + 1 ) sech 2 λ + 4 T [ e 2 κ t + n ¯ ( 1 2 T ) ] tanh 2 λ ,
P = 4 e 2 κ t [ ( a 0 + b 0 ) α 2 2 c 0 tanh λ Re α 2 ] + c 0 ( a 0 b 0 ) ,
a 0 = ( 2 T n ¯ + 1 ) 2 , b 0 = ( 2 T + 2 T n ¯ 1 ) 2 tanh 2 λ ,
c 0 = 4 T 2 n ¯ ( n ¯ + 1 ) + 2 T 1 , T = 1 e 2 κ t .
W m = 1 , n ¯ = 0 ( α , α * , 0 ) = 4 α cosh λ + α * sinh λ 2 1 π e 2 α cosh λ + α * sinh λ 2 ,
P α = 0 = c 0 ( a 0 b 0 ) , ( a 0 b 0 > 0 ) ,
κ t < κ t c 1 2 ln 2 n ¯ + 2 2 n ¯ + 1 ,
P κ t = κ t c = 4 e 2 κ t c ( a 0 + b 0 ) α 2 ,
W m = 1 ( α , α * , ) = 1 π ( 2 n ¯ + 1 ) e [ 2 α 2 ( 2 n ¯ + 1 ) ] ,
p ( n ) = ξ n ( ξ 1 ) n d 2 β π e ξ β 2 L n ( β 2 ) Q ( 1 ξ β ) ,
Q λ , m ( β , t ) = 1 ξ β ρ λ , m ( t ) 1 ξ β = 2 m C λ , m ( t ) m γ m e ( 1 ξ ) β 2 1 γ 2 exp [ ( 1 ξ ) β * 2 + β 2 2 γ β 2 2 ( 1 γ 2 ) e 2 κ t tanh λ ] γ = T tanh λ .
L n ( β β * ) = ( ) n n ! H n , n ( β , β * ) = ( ) n n ! n + n t n t n exp [ t t + t β + t β * ] t = t = 0 ,
p ( n , t ) = ξ n 2 m C λ , m ( t ) m γ m ( L 2 4 I 2 M ) n ( 1 γ 2 ) M P n ( L L 2 4 I 2 ) γ = T tanh λ ,
I = e 2 κ t tanh λ 2 ( 1 γ 2 ) , L = 2 I [ γ + ( 1 ξ ) e 2 κ t tanh λ ] ,
M = 1 + 2 I ( 1 ξ ) [ ( 1 ξ ) e 2 κ t tanh λ + 2 γ ] .
p ( n , t ) ξ 1 = 2 m C λ , m ( t ) e 2 n κ t m γ m ( i tanh λ ) n ( 1 γ 2 ) n 2 + 1 2 P n ( i γ 1 γ 2 ) γ = T tanh λ ,
p ( n , 0 ) ξ 1 = { ( 2 k ) ! 2 2 k k ! k ! sech λ tanh 2 k λ , n = 2 k 0 n = 2 k + 1 } ,
p ( n , 0 ) ξ 1 = { ( 2 k + 1 ) ! 2 2 k 1 k ! k ! sech λ tanh 2 k + 1 λ sinh 2 λ n = 2 k + 1 0 n = 2 k } ,
R ( q ) f , g = δ ( q f q g p ) Tr [ Δ ( β ) ρ ( t ) ] d q d p = Tr [ q f , g q f , g ρ ( t ) ] = q f , g ρ ( t ) q f , g ,
q f , g = A exp [ 2 q a B B * 2 B a 2 ] 0 ,
R ( q , t ) f , g = γ 2 C λ , 1 ( t ) 1 γ 2 q f , g : exp [ a a e 2 κ t tanh λ 2 ( 1 γ 2 ) ( a 2 + a 2 2 γ a a ) ] : q f , g γ = T tanh λ .
R ( q , 0 ) f , g = N λ , 1 2 q f , g a S ( λ ) 0 2 = 2 q 2 A 2 B 2 tanh 3 λ 1 B B * tanh λ 3 exp [ 2 K q 2 1 B B * tanh λ 2 ] ,
R ( q , t ) f , g = A 2 cosh 3 λ 2 F q 2 + E T E 5 2 e ( 2 K q 2 E ) e 2 κ t ,
R ( q , ) f , g = [ π ( f 2 + g 2 ) ] 1 4 exp [ q 2 2 ( f 2 + g 2 ) ] ,
P m ( x ) = l = 0 [ m 2 ] ( 1 ) l ( 2 m 2 l ) ! 2 m l ! ( m l ) ! ( m 2 l ) ! x m 2 l ,
m = 0 , P 0 ( x ) = 1 ,
m = 1 , P 1 ( x ) = x ,
m = 2 , P 2 ( x ) = x 2 [ 1 + 1 2 ( 1 1 x 2 ) ] = 3 2 x 2 1 2 ,
m = 3 , P 3 ( x ) = x 3 [ 1 + 3 2 ( 1 1 x 2 ) ] = 5 2 x 3 3 2 x ,
m = 4 ,
P 4 ( x ) = x 4 [ 1 + 3 ( 1 1 x 2 ) + 3 8 ( 1 1 x 2 ) 2 ] = 1 8 ( 35 x 4 30 x 2 + 3 ) ,
m = 5 ,
P 5 ( x ) = x 5 [ 1 + 5 ( 1 1 x 2 ) + 15 8 ( 1 1 x 2 ) 2 ] = 1 8 ( 63 x 5 70 x 3 + 15 x ) ,
m = 6 ,
P 6 ( x ) = x 6 [ 1 + 15 2 ( 1 1 x 2 ) + 45 8 ( 1 1 x 2 ) 2 + 5 16 ( 1 1 x 2 ) 3 ] = 1 16 ( 231 x 6 315 x 4 + 105 x 2 5 ) ,
m = 7 ,
P 7 ( x ) = x 7 [ 1 + 21 2 ( 1 1 x 2 ) + 105 8 ( 1 1 x 2 ) 2 + 35 16 ( 1 1 x 2 ) 3 ] = 1 16 ( 429 x 7 693 x 5 + 315 x 3 35 x ) .
H m ( x ) = 2 m π ( x + i t ) m e t 2 d t ,
k = 0 z k k ! H m + k ( x ) H m + k ( y ) = 1 π m z m exp [ 4 z ( x + i t ) ( y + i τ ) τ 2 t 2 ] d t d τ .
exp [ α x 2 + β x ] d x = π α exp ( β 2 4 α ) , Re α > 0 ,
k = 0 z k k ! H m + k ( x ) H m + k ( y ) = m z m 1 1 4 z 2 exp [ 4 z 1 4 z 2 ( z x 2 x y + z y 2 ) ] .
p ( n , t ) = ( ) n n ! n + n t n t n e t t d 2 β π exp { [ 1 + 2 γ ( 1 ξ ) I ] β 2 } × exp [ t β + t β * + I ( 1 ξ ) ( β * 2 + β 2 ) ] t = t = 0 = ( ) n n ! 1 M n + n t n t n exp { ( 1 ξ ) [ L M t t + I M ( t 2 + t 2 ) ] } t = t = 0 = L n M n + 1 2 j = 0 [ n 2 ] ( ξ 1 ) n n ! 2 2 j ( n 2 j ) ! j ! j ! ( 2 I L ) 2 j ,
p ( n , t ) = ( ξ 1 ) n M n + 1 2 ( L 2 4 I 2 ) n P n ( L L 2 4 I 2 ) .
p ( n , t ) = ξ n 2 m C λ , m ( t ) ( ξ 1 ) n m γ m p ( n , t ) 1 γ 2 = Eq. ( 42 ) .

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