Abstract

We study nonclassical properties of the optical field when photon subtraction/addition coherent operation ta+ra (|t|2+|r|2=1) acts on a squeezed thermal state (STS) by examining its quadrature squeezing, sub- Poissonian statistics, and negativity of the Wigner function. The degree of squeezing of the coherent operated squeezed thermal state (COSTS) becomes weaker. The Mandel Q parameter decreases with r and quickly becomes negative, which implies sub-Poissonian statistics. Both the negative dip and negative area of the Wigner function increase with r for small squeezing values of κ. Decoherence of the COSTS in an amplitude-damping channel is studied by the time evolution of the Wigner function. The length of time that this nonclassical field preserves its partial negativity of the Wigner function can be modulated by the coherent operation. All these results indicate that the nonclassicality of the STS is sensitive to the coherent operation.

© 2011 Optical Society of America

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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  12. V. Pariga, A. Zavatta, M. Kim, and M. Bellini, “Probing quantum commutation rules by addition and subtraction of single photons to/from a light field,” Science 317, 1890–1893 (2007).
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    [Crossref]
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2010 (7)

F. Dell’Anno, S. De Siena, and F. Illuminati, “Realistic continuous-variable quantum teleportation with non-Gaussian resources,” Phys. Rev. A 81, 012333–012344 (2010).
[Crossref]

F. Dell’Anno, S. De Siena, G. Adesso, and F. Illuminati, “Teleportation of squeezing: optimization using non-Gaussian resources,” Phys. Rev. A 82, 062329–062337 (2010).
[Crossref]

L. Y. Hu, X. X. Xu, and H. Y. Fan, “Statistical properties of photon-subtracted two-mode squeezed vacuum and its decoherence in thermal environment,” J. Opt. Soc. Am. B 27, 286–299(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, 043842–043853 (2010).
[Crossref]

X. X. Xu, L. Y. Hu, and H. Y. Fan, “Photon-added squeezed thermal states: statistical properties and its decoherence in a photon-loss channel,” Opt. Commun. 283, 1801–1809 (2010).
[Crossref]

H. Takahashi, J. S. Neergaard-Nielsen, M. Takeuchi, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Entanglement distillation from Gaussian input states,” Nat. Photonics 4, 178–181(2010).
[Crossref]

S. Y. Lee and H. Nha, “Engineering quantum operations on traveling light beams by multiple photon addition and subtraction,” Phys. Rev. A 82, 053812–053818 (2010).
[Crossref]

2009 (5)

J. Fiurášek, “Engineering quantum operations on traveling light beams by multiple photon addition and subtraction,” Phys. Rev. A 80, 053822–053828 (2009).
[Crossref]

A. Zavatta, V. Parigi, M. S. Kim, H. Jeong, and M. Bellini, “Experimental demonstration of the bosonic commutation relation via superpositions of quantum operations on thermal light fields,” Phys. Rev. Lett. 103, 140406–140409 (2009).
[Crossref] [PubMed]

J. Lee, J. Kim, and H. Nha, “Demonstrating higher-order nonclassical effects by photon-added classical states: realistic schemes,” J. Opt. Soc. Am. B 26, 1363–1369 (2009).
[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, 022315–022323 (2009).
[Crossref]

L. Y. Hu and H. Y. Fan, “Time evolution of Wigner function in laser process derived by entangled state representation,” Opt. Commun. 282, 4379–4383 (2009).
[Crossref]

2008 (5)

M. S. Kim, “Recent developments in photon-level operations on travelling light fields,” J. Phys. B 41, 133001 (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, T. T. Wang, and L. Y. Hu, “Normally ordered bivariate-normal-distribution forms of two-mode mixed states with entanglement involved,” Chin. Phys. Lett. 25, 3539–3542 (2008).
[Crossref]

H. Y. Fan, “Newton–Leibniz integration for ket-bra operators in quantum mechanics (V)—deriving normally ordered bivariate-normal-distribution form of density operators and developing their phase space formalism,” Ann. Phys. 323, 1502–1528 (2008).
[Crossref]

M. S. Kim, H. Jeong, A. Zavatta, V. Parigi, and M. Bellini, “Scheme for proving the bosonic commutation relation using single-photon interference,” Phys. Rev. Lett. 101, 260401–260404 (2008).
[Crossref]

2007 (5)

R. W. Boyd, K. W. Chan, and M. N. O’Sullivan, “Quantum weirdness in the lab,” Science 317, 1874–1875 (2007).
[Crossref] [PubMed]

A. Zavatta, V. Parigi, and M. Bellini, “Experimental nonclassicality of single-photon-added thermal light states,” Phys. Rev. A 75, 052106–052111 (2007).
[Crossref]

V. Pariga, A. Zavatta, M. Kim, and M. Bellini, “Probing quantum commutation rules by addition and subtraction of single photons to/from a light field,” Science 317, 1890–1893 (2007).
[Crossref]

A. Biswas and G. S. Agarwal, “Nonclassicality and decoherence of photon-subtracted squeezed states,” Phys. Rev. A 75, 032104–032111 (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, 022301–022311 (2007).
[Crossref]

2006 (3)

F. Dell’Anno, S. De Siena, and F. Illuminati, “Multiphoton quantum optics and quantum state engineering,” Phys. Rep. 428, 53–168 (2006), and references therein.
[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, 042310–042321 (2006).
[Crossref]

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

2005 (6)

V. V. Dodonov and L. A. de Souza, “Decoherence of superpositions of displaced number states,” J. Opt. B: Quantum Semiclass. Opt. 7, S490–S499 (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, 070501–070504 (2005).
[Crossref] [PubMed]

A. Kitagawa, M. Takeoka, K. Wakui, and M. Sasaki, “Effective squeezing enhancement via measurement-induced non-Gaussian operation and its application to the dense coding scheme,” Phys. Rev. A 72, 022334–022344 (2005).
[Crossref]

S. Olivares and M. G. A. Paris, “Squeezed Fock state by inconclusive photon subtraction,” J. Opt. B: Quantum Semiclass. 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 Semiclass. Opt. 7, S392–S397 (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)

J. Wenger, R. Tualle-Brouri, and P. Grangier, “Non-Gaussian statistics from individual pulses of squeezed light,” Phys. Rev. Lett. 92, 153601–153604 (2004).
[Crossref] [PubMed]

H. Nha and H. J. Carmichael, “Proposed test of quantum nonlocality for continuous variables,” Phys. Rev. Lett. 93, 020401–020404 (2004).
[Crossref] [PubMed]

2002 (2)

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

M. S. Kim, W. Son, V. Bušek, and P. L. Knight, “Entanglement by a beam splitter: nonclassicality as a prerequisite for entanglement,” Phys. Rev. A 65, 032323–032329 (2002).
[Crossref]

1993 (1)

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

1992 (1)

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

1991 (1)

G. S. Agarwal and K. Tara, “Nonclassical properties of states generated by the excitations on a coherent state,” Phys. Rev. A 43, 492–497 (1991).
[Crossref] [PubMed]

1988 (1)

H. Fearn and M. J. Colletta, “Representations of squeezed states with thermal noise,” J. Mod. Opt. 35, 553–564 (1988).
[Crossref]

1987 (1)

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

1963 (1)

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

1932 (1)

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

Adesso, G.

F. Dell’Anno, S. De Siena, G. Adesso, and F. Illuminati, “Teleportation of squeezing: optimization using non-Gaussian resources,” Phys. Rev. A 82, 062329–062337 (2010).
[Crossref]

Agarwal, G. S.

A. Biswas and G. S. Agarwal, “Nonclassicality and decoherence of photon-subtracted squeezed states,” Phys. Rev. A 75, 032104–032111 (2007).
[Crossref]

G. S. Agarwal and K. Tara, “Nonclassical properties of states generated by the excitations on a coherent state,” Phys. Rev. A 43, 492–497 (1991).
[Crossref] [PubMed]

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, 022301–022311 (2007).
[Crossref]

Bellini, M.

A. Zavatta, V. Parigi, M. S. Kim, H. Jeong, and M. Bellini, “Experimental demonstration of the bosonic commutation relation via superpositions of quantum operations on thermal light fields,” Phys. Rev. Lett. 103, 140406–140409 (2009).
[Crossref] [PubMed]

M. S. Kim, H. Jeong, A. Zavatta, V. Parigi, and M. Bellini, “Scheme for proving the bosonic commutation relation using single-photon interference,” Phys. Rev. Lett. 101, 260401–260404 (2008).
[Crossref]

V. Pariga, A. Zavatta, M. Kim, and M. Bellini, “Probing quantum commutation rules by addition and subtraction of single photons to/from a light field,” Science 317, 1890–1893 (2007).
[Crossref]

A. Zavatta, V. Parigi, and M. Bellini, “Experimental nonclassicality of single-photon-added thermal light states,” Phys. Rev. A 75, 052106–052111 (2007).
[Crossref]

Biswas, A.

A. Biswas and G. S. Agarwal, “Nonclassicality and decoherence of photon-subtracted squeezed states,” Phys. Rev. A 75, 032104–032111 (2007).
[Crossref]

Boyd, R. W.

R. W. Boyd, K. W. Chan, and M. N. O’Sullivan, “Quantum weirdness in the lab,” Science 317, 1874–1875 (2007).
[Crossref] [PubMed]

Bušek, V.

M. S. Kim, W. Son, V. Bušek, and P. L. Knight, “Entanglement by a beam splitter: nonclassicality as a prerequisite for entanglement,” Phys. Rev. A 65, 032323–032329 (2002).
[Crossref]

Carmichael, H. J.

H. Nha and H. J. Carmichael, “Proposed test of quantum nonlocality for continuous variables,” Phys. Rev. Lett. 93, 020401–020404 (2004).
[Crossref] [PubMed]

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, 070501–070504 (2005).
[Crossref] [PubMed]

Chan, K. W.

R. W. Boyd, K. W. Chan, and M. N. O’Sullivan, “Quantum weirdness in the lab,” Science 317, 1874–1875 (2007).
[Crossref] [PubMed]

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, 042310–042321 (2006).
[Crossref]

Colletta, M. J.

H. Fearn and M. J. Colletta, “Representations of squeezed states with thermal noise,” J. Mod. Opt. 35, 553–564 (1988).
[Crossref]

De Siena, S.

F. Dell’Anno, S. De Siena, and F. Illuminati, “Realistic continuous-variable quantum teleportation with non-Gaussian resources,” Phys. Rev. A 81, 012333–012344 (2010).
[Crossref]

F. Dell’Anno, S. De Siena, G. Adesso, and F. Illuminati, “Teleportation of squeezing: optimization using non-Gaussian resources,” Phys. Rev. A 82, 062329–062337 (2010).
[Crossref]

F. Dell’Anno, S. De Siena, L. Albano, and F. Illuminati, “Continuous-variable quantum teleportation with non-Gaussian resources,” Phys. Rev. A 76, 022301–022311 (2007).
[Crossref]

F. Dell’Anno, S. De Siena, and F. Illuminati, “Multiphoton quantum optics and quantum state engineering,” Phys. Rep. 428, 53–168 (2006), and references therein.
[Crossref]

de Souza, L. A.

V. V. Dodonov and L. A. de Souza, “Decoherence of superpositions of displaced number states,” J. Opt. B: Quantum Semiclass. Opt. 7, S490–S499 (2005).
[Crossref]

Dell’Anno, F.

F. Dell’Anno, S. De Siena, G. Adesso, and F. Illuminati, “Teleportation of squeezing: optimization using non-Gaussian resources,” Phys. Rev. A 82, 062329–062337 (2010).
[Crossref]

F. Dell’Anno, S. De Siena, and F. Illuminati, “Realistic continuous-variable quantum teleportation with non-Gaussian resources,” Phys. Rev. A 81, 012333–012344 (2010).
[Crossref]

F. Dell’Anno, S. De Siena, L. Albano, and F. Illuminati, “Continuous-variable quantum teleportation with non-Gaussian resources,” Phys. Rev. A 76, 022301–022311 (2007).
[Crossref]

F. Dell’Anno, S. De Siena, and F. Illuminati, “Multiphoton quantum optics and quantum state engineering,” Phys. Rep. 428, 53–168 (2006), and references therein.
[Crossref]

Dodonov, V. V.

V. V. Dodonov and L. A. de Souza, “Decoherence of superpositions of displaced number states,” J. Opt. B: Quantum Semiclass. Opt. 7, S490–S499 (2005).
[Crossref]

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

Fan, H. Y.

X. X. Xu, L. Y. Hu, and H. Y. Fan, “Photon-added squeezed thermal states: statistical properties and its decoherence in a photon-loss channel,” Opt. Commun. 283, 1801–1809 (2010).
[Crossref]

L. Y. Hu, X. X. Xu, and H. Y. Fan, “Statistical properties of photon-subtracted two-mode squeezed vacuum and its decoherence in thermal environment,” J. Opt. Soc. Am. B 27, 286–299(2010).
[Crossref]

L. Y. Hu and H. Y. Fan, “Time evolution of Wigner function in laser process derived by entangled state representation,” Opt. Commun. 282, 4379–4383 (2009).
[Crossref]

H. Y. Fan, T. T. Wang, and L. Y. Hu, “Normally ordered bivariate-normal-distribution forms of two-mode mixed states with entanglement involved,” Chin. Phys. Lett. 25, 3539–3542 (2008).
[Crossref]

H. Y. Fan, “Newton–Leibniz integration for ket-bra operators in quantum mechanics (V)—deriving normally ordered bivariate-normal-distribution form of density operators and developing their phase space formalism,” Ann. Phys. 323, 1502–1528 (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 and H. R. Zaidi, “Application of IWOP technique to the generalized Weyl correspondence,” Phys. Lett. A 124, 303–307(1987).
[Crossref]

Fearn, H.

H. Fearn and M. J. Colletta, “Representations of squeezed states with thermal noise,” J. Mod. Opt. 35, 553–564 (1988).
[Crossref]

Fiurášek, J.

J. Fiurášek, “Engineering quantum operations on traveling light beams by multiple photon addition and subtraction,” Phys. Rev. A 80, 053822–053828 (2009).
[Crossref]

Furusawa, A.

H. Takahashi, J. S. Neergaard-Nielsen, M. Takeuchi, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Entanglement distillation from Gaussian input states,” Nat. Photonics 4, 178–181(2010).
[Crossref]

Glauber, R.

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

Grangier, P.

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

J. Wenger, R. Tualle-Brouri, and P. Grangier, “Non-Gaussian statistics from individual pulses of squeezed light,” Phys. Rev. Lett. 92, 153601–153604 (2004).
[Crossref] [PubMed]

Hayasaka, K.

H. Takahashi, J. S. Neergaard-Nielsen, M. Takeuchi, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Entanglement distillation from Gaussian input states,” Nat. Photonics 4, 178–181(2010).
[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, 043842–043853 (2010).
[Crossref]

L. Y. Hu, X. X. Xu, and H. Y. Fan, “Statistical properties of photon-subtracted two-mode squeezed vacuum and its decoherence in thermal environment,” J. Opt. Soc. Am. B 27, 286–299(2010).
[Crossref]

X. X. Xu, L. Y. Hu, and H. Y. Fan, “Photon-added squeezed thermal states: statistical properties and its decoherence in a photon-loss channel,” Opt. Commun. 283, 1801–1809 (2010).
[Crossref]

L. Y. Hu and H. Y. Fan, “Time evolution of Wigner function in laser process derived by entangled state representation,” Opt. Commun. 282, 4379–4383 (2009).
[Crossref]

H. Y. Fan, T. T. Wang, and L. Y. Hu, “Normally ordered bivariate-normal-distribution forms of two-mode mixed states with entanglement involved,” Chin. Phys. Lett. 25, 3539–3542 (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]

Illuminati, F.

F. Dell’Anno, S. De Siena, G. Adesso, and F. Illuminati, “Teleportation of squeezing: optimization using non-Gaussian resources,” Phys. Rev. A 82, 062329–062337 (2010).
[Crossref]

F. Dell’Anno, S. De Siena, and F. Illuminati, “Realistic continuous-variable quantum teleportation with non-Gaussian resources,” Phys. Rev. A 81, 012333–012344 (2010).
[Crossref]

F. Dell’Anno, S. De Siena, L. Albano, and F. Illuminati, “Continuous-variable quantum teleportation with non-Gaussian resources,” Phys. Rev. A 76, 022301–022311 (2007).
[Crossref]

F. Dell’Anno, S. De Siena, and F. Illuminati, “Multiphoton quantum optics and quantum state engineering,” Phys. Rep. 428, 53–168 (2006), and references therein.
[Crossref]

Jeong, H.

A. Zavatta, V. Parigi, M. S. Kim, H. Jeong, and M. Bellini, “Experimental demonstration of the bosonic commutation relation via superpositions of quantum operations on thermal light fields,” Phys. Rev. Lett. 103, 140406–140409 (2009).
[Crossref] [PubMed]

M. S. Kim, H. Jeong, A. Zavatta, V. Parigi, and M. Bellini, “Scheme for proving the bosonic commutation relation using single-photon interference,” Phys. Rev. Lett. 101, 260401–260404 (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]

Kim, J.

Kim, M.

V. Pariga, A. Zavatta, M. Kim, and M. Bellini, “Probing quantum commutation rules by addition and subtraction of single photons to/from a light field,” Science 317, 1890–1893 (2007).
[Crossref]

Kim, M. S.

A. Zavatta, V. Parigi, M. S. Kim, H. Jeong, and M. Bellini, “Experimental demonstration of the bosonic commutation relation via superpositions of quantum operations on thermal light fields,” Phys. Rev. Lett. 103, 140406–140409 (2009).
[Crossref] [PubMed]

M. S. Kim, H. Jeong, A. Zavatta, V. Parigi, and M. Bellini, “Scheme for proving the bosonic commutation relation using single-photon interference,” Phys. Rev. Lett. 101, 260401–260404 (2008).
[Crossref]

M. S. Kim, “Recent developments in photon-level operations on travelling light fields,” J. Phys. B 41, 133001 (2008).
[Crossref]

M. S. Kim, W. Son, V. Bušek, and P. L. Knight, “Entanglement by a beam splitter: nonclassicality as a prerequisite for entanglement,” Phys. Rev. A 65, 032323–032329 (2002).
[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, 042310–042321 (2006).
[Crossref]

A. Kitagawa, M. Takeoka, K. Wakui, and M. Sasaki, “Effective squeezing enhancement via measurement-induced non-Gaussian operation and its application to the dense coding scheme,” Phys. Rev. A 72, 022334–022344 (2005).
[Crossref]

Knight, P. L.

M. S. Kim, W. Son, V. Bušek, and P. L. Knight, “Entanglement by a beam splitter: nonclassicality as a prerequisite for entanglement,” Phys. Rev. A 65, 032323–032329 (2002).
[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, 070501–070504 (2005).
[Crossref] [PubMed]

Laurat, J.

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

Lee, J.

Lee, S. Y.

S. Y. Lee and H. Nha, “Engineering quantum operations on traveling light beams by multiple photon addition and subtraction,” Phys. Rev. A 82, 053812–053818 (2010).
[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, 022315–022323 (2009).
[Crossref]

Louisell, W. H.

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

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]

Marian, P.

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

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

Marian, T. A.

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

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, 070501–070504 (2005).
[Crossref] [PubMed]

Neergaard-Nielsen, J. S.

H. Takahashi, J. S. Neergaard-Nielsen, M. Takeuchi, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Entanglement distillation from Gaussian input states,” Nat. Photonics 4, 178–181(2010).
[Crossref]

Nha, H.

S. Y. Lee and H. Nha, “Engineering quantum operations on traveling light beams by multiple photon addition and subtraction,” Phys. Rev. A 82, 053812–053818 (2010).
[Crossref]

J. Lee, J. Kim, and H. Nha, “Demonstrating higher-order nonclassical effects by photon-added classical states: realistic schemes,” J. Opt. Soc. Am. B 26, 1363–1369 (2009).
[Crossref]

H. Nha and H. J. Carmichael, “Proposed test of quantum nonlocality for continuous variables,” Phys. Rev. Lett. 93, 020401–020404 (2004).
[Crossref] [PubMed]

O’Sullivan, M. N.

R. W. Boyd, K. W. Chan, and M. N. O’Sullivan, “Quantum weirdness in the lab,” Science 317, 1874–1875 (2007).
[Crossref] [PubMed]

Olivares, S.

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

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

Ourjoumtsev, A.

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

Pariga, V.

V. Pariga, A. Zavatta, M. Kim, and M. Bellini, “Probing quantum commutation rules by addition and subtraction of single photons to/from a light field,” Science 317, 1890–1893 (2007).
[Crossref]

Parigi, V.

A. Zavatta, V. Parigi, M. S. Kim, H. Jeong, and M. Bellini, “Experimental demonstration of the bosonic commutation relation via superpositions of quantum operations on thermal light fields,” Phys. Rev. Lett. 103, 140406–140409 (2009).
[Crossref] [PubMed]

M. S. Kim, H. Jeong, A. Zavatta, V. Parigi, and M. Bellini, “Scheme for proving the bosonic commutation relation using single-photon interference,” Phys. Rev. Lett. 101, 260401–260404 (2008).
[Crossref]

A. Zavatta, V. Parigi, and M. Bellini, “Experimental nonclassicality of single-photon-added thermal light states,” Phys. Rev. A 75, 052106–052111 (2007).
[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 (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 Semiclass. Opt. 7, S392–S397 (2005).
[Crossref]

Puri, R. R.

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

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]

Sasaki, M.

H. Takahashi, J. S. Neergaard-Nielsen, M. Takeuchi, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Entanglement distillation from Gaussian input states,” Nat. Photonics 4, 178–181(2010).
[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, 042310–042321 (2006).
[Crossref]

A. Kitagawa, M. Takeoka, K. Wakui, and M. Sasaki, “Effective squeezing enhancement via measurement-induced non-Gaussian operation and its application to the dense coding scheme,” Phys. Rev. A 72, 022334–022344 (2005).
[Crossref]

Scully, M. O.

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University, 1998).

Son, W.

M. S. Kim, W. Son, V. Bušek, and P. L. Knight, “Entanglement by a beam splitter: nonclassicality as a prerequisite for entanglement,” Phys. Rev. A 65, 032323–032329 (2002).
[Crossref]

Takahashi, H.

H. Takahashi, J. S. Neergaard-Nielsen, M. Takeuchi, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Entanglement distillation from Gaussian input states,” Nat. Photonics 4, 178–181(2010).
[Crossref]

Takeoka, M.

H. Takahashi, J. S. Neergaard-Nielsen, M. Takeuchi, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Entanglement distillation from Gaussian input states,” Nat. Photonics 4, 178–181(2010).
[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, 042310–042321 (2006).
[Crossref]

A. Kitagawa, M. Takeoka, K. Wakui, and M. Sasaki, “Effective squeezing enhancement via measurement-induced non-Gaussian operation and its application to the dense coding scheme,” Phys. Rev. A 72, 022334–022344 (2005).
[Crossref]

Takeuchi, M.

H. Takahashi, J. S. Neergaard-Nielsen, M. Takeuchi, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Entanglement distillation from Gaussian input states,” Nat. Photonics 4, 178–181(2010).
[Crossref]

Tara, K.

G. S. Agarwal and K. Tara, “Nonclassical properties of states generated by the excitations on a coherent state,” Phys. Rev. A 43, 492–497 (1991).
[Crossref] [PubMed]

Tualle-Brouri, R.

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

J. Wenger, R. Tualle-Brouri, and P. Grangier, “Non-Gaussian statistics from individual pulses of squeezed light,” Phys. Rev. Lett. 92, 153601–153604 (2004).
[Crossref] [PubMed]

Wakui, K.

A. Kitagawa, M. Takeoka, K. Wakui, and M. Sasaki, “Effective squeezing enhancement via measurement-induced non-Gaussian operation and its application to the dense coding scheme,” Phys. Rev. A 72, 022334–022344 (2005).
[Crossref]

Wang, T. T.

H. Y. Fan, T. T. Wang, and L. Y. Hu, “Normally ordered bivariate-normal-distribution forms of two-mode mixed states with entanglement involved,” Chin. Phys. Lett. 25, 3539–3542 (2008).
[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, 043842–043853 (2010).
[Crossref]

Wenger, J.

J. Wenger, R. Tualle-Brouri, and P. Grangier, “Non-Gaussian statistics from individual pulses of squeezed light,” Phys. Rev. Lett. 92, 153601–153604 (2004).
[Crossref] [PubMed]

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, 070501–070504 (2005).
[Crossref] [PubMed]

Wigner, E.

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[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, 070501–070504 (2005).
[Crossref] [PubMed]

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, 043842–043853 (2010).
[Crossref]

Xu, X. X.

X. X. Xu, L. Y. Hu, and H. Y. Fan, “Photon-added squeezed thermal states: statistical properties and its decoherence in a photon-loss channel,” Opt. Commun. 283, 1801–1809 (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, 043842–043853 (2010).
[Crossref]

L. Y. Hu, X. X. Xu, and H. Y. Fan, “Statistical properties of photon-subtracted two-mode squeezed vacuum and its decoherence in thermal environment,” J. Opt. Soc. Am. B 27, 286–299(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, 022315–022323 (2009).
[Crossref]

Zaidi, H. R.

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

Zavatta, A.

A. Zavatta, V. Parigi, M. S. Kim, H. Jeong, and M. Bellini, “Experimental demonstration of the bosonic commutation relation via superpositions of quantum operations on thermal light fields,” Phys. Rev. Lett. 103, 140406–140409 (2009).
[Crossref] [PubMed]

M. S. Kim, H. Jeong, A. Zavatta, V. Parigi, and M. Bellini, “Scheme for proving the bosonic commutation relation using single-photon interference,” Phys. Rev. Lett. 101, 260401–260404 (2008).
[Crossref]

A. Zavatta, V. Parigi, and M. Bellini, “Experimental nonclassicality of single-photon-added thermal light states,” Phys. Rev. A 75, 052106–052111 (2007).
[Crossref]

V. Pariga, A. Zavatta, M. Kim, and M. Bellini, “Probing quantum commutation rules by addition and subtraction of single photons to/from a light field,” Science 317, 1890–1893 (2007).
[Crossref]

Zubairy, M. S.

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University, 1998).

Ann. Phys. (1)

H. Y. Fan, “Newton–Leibniz integration for ket-bra operators in quantum mechanics (V)—deriving normally ordered bivariate-normal-distribution form of density operators and developing their phase space formalism,” Ann. Phys. 323, 1502–1528 (2008).
[Crossref]

Chin. Phys. Lett. (1)

H. Y. Fan, T. T. Wang, and L. Y. Hu, “Normally ordered bivariate-normal-distribution forms of two-mode mixed states with entanglement involved,” Chin. Phys. Lett. 25, 3539–3542 (2008).
[Crossref]

J. Mod. Opt. (1)

H. Fearn and M. J. Colletta, “Representations of squeezed states with thermal noise,” J. Mod. Opt. 35, 553–564 (1988).
[Crossref]

J. Opt. B: Quantum Semiclass. Opt. (4)

V. V. Dodonov and L. A. de Souza, “Decoherence of superpositions of displaced number states,” J. Opt. B: Quantum Semiclass. Opt. 7, S490–S499 (2005).
[Crossref]

V. V. Dodonov, “Review Article: ‘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 (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 Semiclass. Opt. 7, S392–S397 (2005).
[Crossref]

J. Opt. Soc. Am. B (3)

J. Phys. B (1)

M. S. Kim, “Recent developments in photon-level operations on travelling light fields,” J. Phys. B 41, 133001 (2008).
[Crossref]

Nat. Photonics (1)

H. Takahashi, J. S. Neergaard-Nielsen, M. Takeuchi, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Entanglement distillation from Gaussian input states,” Nat. Photonics 4, 178–181(2010).
[Crossref]

Opt. Commun. (2)

X. X. Xu, L. Y. Hu, and H. Y. Fan, “Photon-added squeezed thermal states: statistical properties and its decoherence in a photon-loss channel,” Opt. Commun. 283, 1801–1809 (2010).
[Crossref]

L. Y. Hu and H. Y. Fan, “Time evolution of Wigner function in laser process derived by entangled state representation,” Opt. Commun. 282, 4379–4383 (2009).
[Crossref]

Phys. Lett. A (1)

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

Phys. Rep. (1)

F. Dell’Anno, S. De Siena, and F. Illuminati, “Multiphoton quantum optics and quantum state engineering,” Phys. Rep. 428, 53–168 (2006), and references therein.
[Crossref]

Phys. Rev. (2)

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

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

Phys. Rev. A (16)

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

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

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]

J. Fiurášek, “Engineering quantum operations on traveling light beams by multiple photon addition and subtraction,” Phys. Rev. A 80, 053822–053828 (2009).
[Crossref]

S. Y. Lee and H. Nha, “Engineering quantum operations on traveling light beams by multiple photon addition and subtraction,” Phys. Rev. A 82, 053812–053818 (2010).
[Crossref]

M. S. Kim, W. Son, V. Bušek, and P. L. Knight, “Entanglement by a beam splitter: nonclassicality as a prerequisite for entanglement,” Phys. Rev. A 65, 032323–032329 (2002).
[Crossref]

A. Kitagawa, M. Takeoka, K. Wakui, and M. Sasaki, “Effective squeezing enhancement via measurement-induced non-Gaussian operation and its application to the dense coding scheme,” Phys. Rev. A 72, 022334–022344 (2005).
[Crossref]

F. Dell’Anno, S. De Siena, L. Albano, and F. Illuminati, “Continuous-variable quantum teleportation with non-Gaussian resources,” Phys. Rev. A 76, 022301–022311 (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, 022315–022323 (2009).
[Crossref]

F. Dell’Anno, S. De Siena, and F. Illuminati, “Realistic continuous-variable quantum teleportation with non-Gaussian resources,” Phys. Rev. A 81, 012333–012344 (2010).
[Crossref]

F. Dell’Anno, S. De Siena, G. Adesso, and F. Illuminati, “Teleportation of squeezing: optimization using non-Gaussian resources,” Phys. Rev. A 82, 062329–062337 (2010).
[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, 042310–042321 (2006).
[Crossref]

A. Zavatta, V. Parigi, and M. Bellini, “Experimental nonclassicality of single-photon-added thermal light states,” Phys. Rev. A 75, 052106–052111 (2007).
[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, 043842–043853 (2010).
[Crossref]

A. Biswas and G. S. Agarwal, “Nonclassicality and decoherence of photon-subtracted squeezed states,” Phys. Rev. A 75, 032104–032111 (2007).
[Crossref]

G. S. Agarwal and K. Tara, “Nonclassical properties of states generated by the excitations on a coherent state,” Phys. Rev. A 43, 492–497 (1991).
[Crossref] [PubMed]

Phys. Rev. Lett. (5)

H. Nha and H. J. Carmichael, “Proposed test of quantum nonlocality for continuous variables,” Phys. Rev. Lett. 93, 020401–020404 (2004).
[Crossref] [PubMed]

J. Wenger, R. Tualle-Brouri, and P. Grangier, “Non-Gaussian statistics from individual pulses of squeezed light,” Phys. Rev. Lett. 92, 153601–153604 (2004).
[Crossref] [PubMed]

A. Zavatta, V. Parigi, M. S. Kim, H. Jeong, and M. Bellini, “Experimental demonstration of the bosonic commutation relation via superpositions of quantum operations on thermal light fields,” Phys. Rev. Lett. 103, 140406–140409 (2009).
[Crossref] [PubMed]

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, 070501–070504 (2005).
[Crossref] [PubMed]

M. S. Kim, H. Jeong, A. Zavatta, V. Parigi, and M. Bellini, “Scheme for proving the bosonic commutation relation using single-photon interference,” Phys. Rev. Lett. 101, 260401–260404 (2008).
[Crossref]

Science (3)

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

V. Pariga, A. Zavatta, M. Kim, and M. Bellini, “Probing quantum commutation rules by addition and subtraction of single photons to/from a light field,” Science 317, 1890–1893 (2007).
[Crossref]

R. W. Boyd, K. W. Chan, and M. N. O’Sullivan, “Quantum weirdness in the lab,” Science 317, 1874–1875 (2007).
[Crossref] [PubMed]

Other (3)

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University, 1998).

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

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

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Figures (7)

Fig. 1
Fig. 1

Degree of quadrature squeezing S for the STS and the COSTS. (a) average photon number n ¯ = 0 ; bottom line, S 0 ; top line, S 1 . (b)  n ¯ = 0.1 , except S 0 ; other lines denote S 1 for different values of r (from bottom to top r = 0 , 0.1, 0.3, 0.5, 0.8, 1).

Fig. 2
Fig. 2

Q parameter of the COSTS changes with r for different values of both parameters n ¯ and κ.

Fig. 3
Fig. 3

Wigner function of COSTS for r = 0.3 and n ¯ = 0.1 with (a)  κ = 0 , (b)  κ = 0.2 , and (c)  κ = 0.6 . (d), (e), and (f) are contour lines with interval 0.05, corresponding to (a), (b), and (c), respectively, where only the negative regions are plotted.

Fig. 4
Fig. 4

Wigner function of COSTS for squeezing parameter κ = 0.6 and n ¯ = 0.1 with (a)  r = 0.1 , (b)  r = 0.5 , and (c)  r = 1.0 . (d), (e) and (f) are contour lines with interval 0.05, corresponding to (a), (b), and (c), respectively, where only the negative regions are plotted.

Fig. 5
Fig. 5

Contribution to Wigner function of COSTS from the “off-diagonal” components a ρ s a + a ρ s a for r = 0.3 and n ¯ = 0.1 with (a)  κ = 0 and (b)  κ = 0.6 .

Fig. 6
Fig. 6

Wigner function of COSTS in amplitude-damping channel for κ = 0.3 , r = 0.3 , n = 0.1 , and different γ t : (a)  γ t = 0.05 , (b)  γ t = 0.2 , (c)  γ t = 0.5 , and (d)  γ t = 0.7 .

Fig. 7
Fig. 7

Wigner function of COSTS in amplitude- damping channel for κ = 0.6 , γ t = 0.3 , n ¯ = 0.1 , and different r: (a)  r = 0.0 , (b)  r = 0.3 , and (c)  r = 1 .

Equations (61)

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ρ = N 1 ( t a + r a ) ρ s ( t * a + r * a ) ,
ρ s = S ( κ ) ρ t h S ( κ ) ,
ρ s = 1 τ 1 τ 2 : exp [ X 2 2 τ 1 2 P 2 2 τ 2 2 ] : ,
τ 1 2 τ 2 2 = ( 2 n ¯ + 1 ) sinh 2 κ ,
τ 1 2 + τ 2 2 = 2 ( n ¯ cosh 2 κ + cosh 2 κ ) ,
τ 1 2 τ 2 2 = n ¯ 2 + ( 2 n ¯ + 1 ) cosh 2 κ .
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 ] ,
N = τ 1 2 + τ 2 2 2 2 | t | 2 + ( τ 1 2 τ 2 2 ) 2 ( t r * + r t * ) + τ 1 2 + τ 2 2 2 | r | 2 .
N = | r | 2 + ( 2 n ¯ + 1 ) sinh 2 κ 2 ( t r * + r t * ) + ( 2 n ¯ + 1 ) cosh 2 κ 1 2 .
S opt : Δ 2 X θ : min = 2 | a 2 a 2 | + 2 a a 2 | a | 2 ,
S 0 = ( 2 n ¯ + 1 ) e 2 κ 1 ,
a = T r { ρ a } = 0
a 2 = 6 A ( B + r 2 ) + 2 ( 8 A 2 + B 2 + B ) r t B + r 2 + 4 A t r ,
a a = ( 3 B + 1 ) ( r 2 + 4 t r A ) + 4 A 2 + 2 B 2 B + r 2 + 4 A t r ,
S 1 2 6 A ( B + r 2 ) + 2 ( 8 A 2 + B 2 + B ) r t B + r 2 + 4 A t r + 2 ( 3 B + 1 ) ( r 2 + 4 t r A ) + 4 A 2 + 2 B 2 B + r 2 + 4 A t r ,
S n ¯ = 0 = 3 e 2 κ 1 ,
S κ = 0 = 2 ( 3 n ¯ + 1 ) r 2 2 ( n ¯ 2 + n ¯ ) + 2 n ¯ 2 n ¯ + r 2 ,
Q = a 2 a 2 a a 2 a a .
N a 2 a 2 = N T r { ρ a 2 a 2 } = t 2 B ( 6 B 2 + 36 A 2 ) + r 2 ( 36 A 2 B + 20 A 2 + 6 B 3 + 10 B 2 + 4 B ) + 24 t r A ( 2 A 2 + 2 B 2 + B ) .
Q = ( 20 A 2 + 10 B 2 + 4 B ) r 2 + 24 A ( 2 A 2 + 2 B 2 + B ) r t + 36 A 2 B + 6 B 3 ( 3 B + 1 ) ( r 2 + 4 A t r ) + 4 A 2 + 2 B 2 ( 3 B + 1 ) ( r 2 + 4 t r A ) + 4 A 2 + 2 B 2 r 2 + B + 4 A t r .
Q κ = 0 = 2 n ¯ 2 ( n ¯ + r 2 ) 2 r 4 ( 1 + n ¯ ) 2 ( n ¯ + r 2 ) ( 2 n ¯ 2 + 3 n ¯ r 2 + r 2 ) ,
Q n ¯ = 0 = 3 cosh 4 κ 6 cosh 2 κ 1 6 cosh 2 κ 2 ,
W ( α ) = 2 e 2 | α | 2 π d 2 z π z | ρ | z exp [ 2 ( z α * z * α ) ] ,
W ( α ) = W 0 ( α ) N ( 2 n ¯ + 1 ) { | t | 2 [ | 2 α * ( n ¯ sinh 2 κ ) + α sinh 2 κ | 2 2 n ¯ + 1 + n ¯ sinh 2 κ ] + | r | 2 [ | 2 α * ( n + cosh 2 κ ) α sinh 2 κ | 2 ( 2 n ¯ + 1 ) 2 n ¯ cosh 2 κ ] + ( r * t + t * r ) N [ ( α 2 + α * 2 ) sinh 2 2 κ + | α | 2 sinh 4 κ 2 n ¯ + 1 sinh 2 κ 2 ] + ( r * t α 2 + t * r α * 2 ) 4 n ¯ ( n ¯ + 1 ) N ( 2 n ¯ + 1 ) } ,
W 0 ( α ) = 2 π ( 2 n ¯ + 1 ) exp [ sinh 2 κ 2 n + 1 ( α * 2 + α 2 ) 2 cosh 2 κ 2 n + 1 | α | 2 ] .
W κ = 0 ( α ) = ( 1 + 2 n ¯ ) W t h ( α ) ( 1 + n ¯ ) ( | r | 2 + n ¯ ) [ n ¯ | t | 2 1 + 2 n ¯ ( 1 + 2 n ¯ ) | r | 2 + n ¯ 2 ( 1 + 2 n ¯ ) 2 ( 1 4 ( 1 + n ¯ ) | α | 2 1 + 2 n ¯ ) + 4 n ¯ ( 1 + n ¯ ) 2 ( 1 + 2 n ¯ ) 3 ( t r * α 2 + t * r α * 2 ) ] ,
W n ¯ = 0 ( α ) = W 0 ( α ) ( 1 + 4 | α * sinh κ α cosh κ | 2 ) ,
C 1 x 2 + C 2 y 2 < C 3 ,
C 1 = 4 e 2 κ r 2 + 2 4 n ¯ ( n ¯ + 1 ) e 4 κ + 1 2 n ¯ + 1 r t + [ sinh 2 κ + 2 ( n ¯ sinh 2 κ ) ] 2 2 n ¯ + 1 , C 2 = 4 e 2 κ r 2 2 4 n ¯ ( n ¯ + 1 ) e 4 κ + 1 2 n ¯ + 1 r t + [ sinh 2 κ 2 ( n ¯ sinh 2 κ ) ] 2 2 n ¯ + 1 , C 3 = ( 2 n ¯ + 1 ) r 2 + r t sinh 2 κ ( n ¯ sinh 2 κ ) .
d ρ d t = γ ( 2 a ρ a a a ρ ρ a a ) ,
W ( z , t ) = 2 T d 2 α π W ( α , 0 ) exp [ 2 | z α e γ t | 2 T ] ,
W ( z , t ) = W 0 ( z , t ) F ( z , t ) ,
W 0 ( z , t ) = 2 / ( 2 n ¯ + 1 ) π T G exp [ Δ 1 | z | 2 + g 2 g 3 2 4 G ( z * 2 + z 2 ) ] ,
N F ( z , t ) = | t | 2 ( χ + | ω | 2 ) + | r | 2 ( χ + | ω | 2 ) + ( r * t + t * r ) ( Δ 2 + 2 μ υ g 2 3 G μ 2 μ * 2 ) + 4 n ¯ ( n ¯ + 1 ) ( 2 n ¯ + 1 ) 2 [ g 2 2 G ( r t * + r * t ) + r t * ξ 2 + r * t ξ * 2 ] ,
g 0 = cosh 2 κ 2 n ¯ + 1 , g 1 = n ¯ sinh 2 κ 2 n ¯ + 1 , g 2 = sinh 2 κ 2 n ¯ + 1 , g 3 = 2 e γ t T , g 1 = 1 g 1 ,
δ = 2 g 0 + g 3 e γ t , G = δ 2 4 g 2 2 , Δ 1 = 2 T + δ g 3 2 4 G , Δ 2 = g 2 2 + δ g 0 g 2 2 G , μ = δ g 2 g 3 z + 2 g 2 2 g 3 z * 4 G , χ = δ ( 4 g 1 2 + g 2 2 ) + 8 g 1 g 2 2 4 G + g 1 , χ = δ ( 4 g 1 2 + g 2 2 ) 8 g 1 g 2 2 4 G g 1 , ω = δ g 3 ( 2 g 1 z + g 2 z * ) + g 2 g 3 ( 4 g 1 z * + 2 g 2 z ) 4 G , ω = δ g 3 ( 2 g 1 z g 2 z * ) + g 2 g 3 ( 4 g 1 z * 2 g 2 z ) 4 G , υ = g 0 g 3 ( δ z * + 2 g 2 z ) 4 G , ξ = g 3 ( δ z * + 2 g 2 z ) 4 G .
| t | 2 χ + | r | 2 χ + ( r * t + t * r ) [ Δ 2 g 2 3 G + 4 n ¯ ( n ¯ + 1 ) g 2 2 ( 2 n ¯ + 1 ) 2 G ] < 0 .
γ t < γ t c = 1 2 ln [ 1 ( 2 n ¯ + 1 ) ( n ¯ sinh 2 κ ) n ¯ cosh 2 κ + sinh 2 κ ] ,
W ( z , ) = 2 π exp [ 2 | z | 2 ] ,
N = T r { | t | 2 a ρ s a } + T r { | r | 2 a ρ s a } + T r { r * t a ρ s a } + T r { r t * a ρ s a } .
z 1 | z 2 = exp [ 1 2 | z 1 | 2 1 2 | z 2 | 2 + z 1 * z 2 ] ,
T r { a ρ s a ^ } = 1 τ 1 τ 2 2 s f d z 1 π exp [ τ 1 2 + τ 2 2 2 τ 1 2 τ 2 2 | z 1 | 2 + ( 1 τ 1 2 + τ 2 2 2 τ 1 2 τ 2 2 ) f z 1 * + ( s + f τ 1 2 τ 2 2 2 τ 1 2 τ 2 2 ) z 1 τ 1 2 τ 2 2 + 4 τ 1 2 τ 2 2 ( z 1 2 + z 1 * 2 ) + τ 1 2 τ 2 2 4 τ 1 2 τ 2 2 f 2 ] s = f = 0 = 2 s f exp [ τ 1 2 + τ 2 2 2 2 s f + τ 1 2 τ 2 2 4 ( f 2 + s 2 ) ] s = f = 0 = τ 1 2 + τ 2 2 2 2
T r { a ^ ρ s a } = 2 s f exp [ τ 1 2 + τ 2 2 2 s f + τ 1 2 τ 2 2 4 ( f 2 + s 2 ) ] s = f = 0 = ( τ 1 2 + τ 2 2 ) 2 ,
T r { r * t a ρ s a + r t * a ρ s a } = τ 1 2 τ 2 2 2 ( r * t + r t * ) .
a 2 = T r { ρ a 2 } .
T r { a ρ s a a 2 } = 4 s f 3 exp [ τ 1 2 + τ 2 2 2 2 s f + τ 1 2 τ 2 2 4 ( f 2 + s 2 ) ] s = f = 0 = 6 A B ,
T r { a ^ ρ s a a ^ 2 } = 4 s 3 f exp [ ( τ 1 2 + τ 2 2 ) 2 f s + τ 1 2 τ 2 2 4 ( f 2 + s 2 ) ] s = f = 0 = 6 A ( B + 1 ) ,
T r { a ρ s a a 2 } = T r { a 2 ρ s a 2 } 2 T r { a ρ s a } = 4 A 2 + 2 B 2 + 2 B ,
T r { a ρ s a a 2 } = 12 A 2 .
a = T r { ρ a } = 0 .
a a = 1 N T r { t 2 a 2 ρ s a 2 + r 2 ( a 2 ρ s a 2 a ρ s a ) + r t ( a 3 ρ s a + a 2 ρ s + a ρ s a 3 + ρ s a 2 ) } .
a 2 a 2 = 1 N T r { t 2 a 3 ρ s a 3 + r 2 a ρ s a a 2 a 2 + r t ( a 3 ρ s a a 2 + a ρ s a a 2 a 2 ) } .
W ( α ) = e 2 | α | 2 | t | 2 N π d 2 z π z | a ρ s a | z e 2 ( z α * z * α ) ,
W + ( α ) = e 2 | α | 2 | r | 2 N π d 2 z π z | a ρ s a | z e 2 ( z α * z * α ) ,
W ( α ) = e 2 | α | 2 N π d 2 z π z | ( t r * ρ s a 2 + t * r ρ s a 2 ) | z e 2 ( z α * z * α ) ,
W ( α ) = e 2 | α | 2 | t | 2 N π τ 1 τ 2 2 f s d 2 z π exp [ | z | 2 2 ( z α * z * α ) ] × d 2 z 1 π exp [ | z 1 | 2 ( z * s ) z 1 + τ 1 2 τ 2 2 4 τ 1 2 τ 2 2 z 1 * 2 ] × d 2 z 2 π exp [ | z 2 | 2 + z 1 * z 2 τ 1 2 + τ 2 2 2 τ 1 2 τ 2 2 z 1 * z 2 + ( z + f ) z 2 * z 2 2 4 τ 1 2 + z 2 2 4 τ 2 2 ] s = f = 0 .
W ( α ) = | t | 2 W 0 ( α ) N ( 2 n + 1 ) [ | α sinh 2 κ + 2 α * ( n sinh 2 κ ) | 2 2 n + 1 + n sinh 2 κ ] ,
W + ( α ) = | r | 2 W 0 ( α ) N ( 2 n ¯ + 1 ) [ | 2 α * ( n + cosh 2 κ ) α sinh 2 κ | 2 2 n ¯ + 1 n ¯ cosh 2 κ ] .
W ( α ) = ( r * t + t * r ) W 0 ( α ) N ( 2 n ¯ + 1 ) [ sinh 2 κ 2 + ( α 2 + α * 2 ) sinh 2 2 κ + | α | 2 sinh 4 κ ( 2 n ¯ + 1 ) ] + ( r * t α 2 + t * r α * 2 ) 4 n ¯ ( n ¯ + 1 ) N ( 2 n ¯ + 1 ) 2 W 0 ( α , α * ) .
W ( z , t ) = 2 | t | 2 T N d 2 α π W 0 ( α ) exp [ 2 | z α e γ t | 2 T ] × [ n ¯ sinh 2 κ 2 n ¯ + 1 + | 2 α * ( n ¯ sinh 2 κ ) + α sinh 2 κ | 2 ( 2 n ¯ + 1 ) 2 ] = 4 | t | 2 T N π ( 2 n ¯ + 1 ) exp [ 2 | z | 2 T ] 2 f s { exp [ f s g 1 ] × d 2 α π exp [ ( 2 g 0 + g 3 e γ t ) | α | 2 + ( 2 f g 1 + s g 2 + g 3 z * ) α + ( 2 s g 1 + f g 2 + g 3 z ) α * + g 2 ( α * 2 + α 2 ) ] } f = s = 0 .
W ( z , t ) = 2 | t | 2 ( χ + | ω | 2 ) N π T G ( 2 n ¯ + 1 ) exp [ Δ 1 | z | 2 + g 2 g 3 2 4 G ( z * 2 + z 2 ) ] .

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